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4.6 more on log and exponential equations

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math260

The document discusses two types of log and exponential equations: 1) Equations that do not require calculators, which involve relating bases on both sides of the equation and can be simplified using properties of logarithms and exponents. 2) Numerical equations that do require calculators. It provides examples of solving each type, including consolidating bases, using logarithmic and exponential properties, and algebraic manipulation to isolate the variable.

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More on Log and Exponential Equations http://www.lahc.edu/math/precalculus/math_260a.html
In this section we continue with log-equations and exp-equations. More on Log and Exponential Equations
In this section we continue with log-equations and exp-equations. They may be grouped in two types. More on Log and Exponential Equations
In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). More on Log and Exponential Equations
In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations
In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators
In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators The Law of Uniqueness of Log and Exp Functions:
In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators If logb(U) = logb(V) then U = V. The Law of Uniqueness of Log and Exp Functions
In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators If logb(U) = logb(V) then U = V. If expb(U) = expb(V) or bU = bV then U = V. The Law of Uniqueness of Log and Exp Functions
In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators Use this law to simplify log or exp equations when both sides can be consolidated into a common base. The Law of Uniqueness of Log and Exp Functions If logb(U) = logb(V) then U = V. If expb(U) = expb(V) or bU = bV then U = V.
In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators Use this law to simplify log or exp equations when both sides can be consolidated into a common base. An example of numbers with common bases are {.., 1/8, ¼, ½, 1, 2, 4, 8,..}. The Law of Uniqueness of Log and Exp Functions If logb(U) = logb(V) then U = V. If expb(U) = expb(V) or bU = bV then U = V.
In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators Use this law to simplify log or exp equations when both sides can be consolidated into a common base. An example of numbers with common bases are {.., 1/8, ¼, ½, 1, 2, 4, 8,..}. All these numbers are base 2 numbers, i.e. they are powers of 2. The Law of Uniqueness of Log and Exp Functions If logb(U) = logb(V) then U = V. If expb(U) = expb(V) or bU = bV then U = V.
Example A: Solve 2*42x – 1 = 81 – 3x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first.
Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first.
Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first.
Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv,
Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x 2*24x – 2 = 23 – 9x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv,
Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x 2*24x – 2 = 23 – 9x 21 + 4x – 2 = 23 – 9x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv,
Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x 2*24x – 2 = 23 – 9x 21 + 4x – 2 = 23 – 9x drop the base 2 1 + 4x – 2 = 3 – 9x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv, then drop the base b and solve U = V.
Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x 2*24x – 2 = 23 – 9x 21 + 4x – 2 = 23 – 9x drop the base 2 1 + 4x – 2 = 3 – 9x 4x – 1 = 3 – 9x 13x = 4 Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv, then drop the base b and solve U = V.
Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x 2*24x – 2 = 23 – 9x 21 + 4x – 2 = 23 – 9x drop the base 2 1 + 4x – 2 = 3 – 9x 4x – 1 = 3 – 9x 13x = 4  x = 4/13 Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv, then drop the base b and solve U = V.
For log-equations of this type, consolidate the logs on each side first. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, Equations That Do Not Need Calculators Example B: Solve log2(x – 1) + log2(x + 3) = 5
Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V,
Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 x2 + 2x – 3 = 32 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 x2 + 2x – 3 = 32 x2 + 2x – 35 = 0 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 x2 + 2x – 3 = 32 x2 + 2x – 35 = 0 (x + 7)(x – 5) = 0 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 x2 + 2x – 3 = 32 x2 + 2x – 35 = 0 (x + 7)(x – 5) = 0 x = -7, x = 5 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 x2 + 2x – 3 = 32 x2 + 2x – 35 = 0 (x + 7)(x – 5) = 0 x = -7, x = 5 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), Equations That Do Not Need Calculators
II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) Combine the logs using quotient rule: log4[ ] = log4(5)(x + 2) (x – 1) II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) Combine the logs using quotient rule: log4[ ] = log4(5) Drop the logs on both sides to get (x + 2) (x – 1) (x + 2) (x – 1) = 5 II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) Combine the logs using quotient rule: log4[ ] = log4(5) Drop the logs on both sides to get x + 2 = 5(x – 1) (x + 2) (x – 1) (x + 2) (x – 1) = 5 II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) Combine the logs using quotient rule: log4[ ] = log4(5) Drop the logs on both sides to get x + 2 = 5(x – 1) x + 2 = 5x – 5 (x + 2) (x – 1) (x + 2) (x – 1) = 5 II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) Combine the logs using quotient rule: log4[ ] = log4(5) Drop the logs on both sides to get x + 2 = 5(x – 1) x + 2 = 5x – 5 7 = 4x 7/4 = x (x + 2) (x – 1) (x + 2) (x – 1) = 5 II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators.
Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section.
Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10.
Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10. We do this by changing the calculation into base e with the following formula.
Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10. We do this by changing the calculation into base e with the following formula. Given x and a base b, we've blog (x) = x,b
Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10. We do this by changing the calculation into base e with the following formula. Given x and a base b, we've blog (x) = x, apply ln( ) to both sides: ln(blog (x)) = ln(x) b b
Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10. We do this by changing the calculation into base e with the following formula. Given x and a base b, we've blog (x) = x, apply ln( ) to both sides: ln(blog (x)) = ln(x) use the power rule logb(x)ln(b) = ln(x) b b
Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10. We do this by changing the calculation into base e with the following formula. Given x and a base b, we've blog (x) = x, apply ln( ) to both sides: ln(blog (x)) = ln(x) use the power rule logb(x)ln(b) = ln(x) Therefore we've the Base-Change Formula logb(x) = b b ln(x) ln(b)
Numerical Equations That Require Calculators To solve equations in bases different from e,
Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first,
Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
Example D: (Base Change) Solve for x if 14 = 3·52x+1. Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Rewrite in log5( ): log5(14/3) = 2x+1 Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Rewrite in log5( ): log5(14/3) = 2x+1 Solve for x: log5(14/3) – 1 = 2x Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Rewrite in log5( ): log5(14/3) = 2x+1 Solve for x: log5(14/3) – 1 = 2x (log5(14/3) – 1) / 2 = x Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Rewrite in log5( ): log5(14/3) = 2x+1 Solve for x: log5(14/3) – 1 = 2x (log5(14/3) – 1) / 2 = x By base-change formula using ln: ( – 1) / 2 = xln(14/3) ln(5) Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Rewrite in log5( ): log5(14/3) = 2x+1 Solve for x: log5(14/3) – 1 = 2x (log5(14/3) – 1) / 2 = x By base-change formula using ln: ( – 1) / 2 = x  -0.0214ln(14/3) ln(5) Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Numerical Equations That Require Calculators
Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: Numerical Equations That Require Calculators
Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) Numerical Equations That Require Calculators
Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) Numerical Equations That Require Calculators
Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Numerical Equations That Require Calculators
Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Solve for x: x – 2log4(7)x = log4(7) + 1 Numerical Equations That Require Calculators
Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Solve for x: x – 2log4(7)x = log4(7) + 1 (1 – 2 log4(7))x = log4(7) + 1 Numerical Equations That Require Calculators
Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Solve for x: x – 2log4(7)x = log4(7) + 1 (1 – 2 log4(7))x = log4(7) + 1 x = Numerical Equations That Require Calculators log4(7) + 1 1 – 2log4(7)
Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Solve for x: x – 2log4(7)x = log4(7) + 1 (1 – 2 log4(7))x = log4(7) + 1 x = Change bases x = Numerical Equations That Require Calculators + 1 1 – ln(7) ln(4) 2ln(7) ln(4) log4(7) + 1 1 – 2log4(7)
Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Solve for x: x – 2log4(7)x = log4(7) + 1 (1 – 2 log4(7))x = log4(7) + 1 x = Change bases x = =  -1.33 Numerical Equations That Require Calculators + 1 1 – ln(7) ln(4) 2ln(7) ln(4) log4(7) + 1 1 – 2log4(7) ln(7) + ln(4) ln(4) – 2ln(7)
More on Log and Exponential Equations
More on Log and Exponential Equations
(Answers to the odd problems) Exercise A. 1. 𝑥 = 2/5 3. 𝑥 = 2/5 5. 𝑥 = 3/10 7. 𝑥 = 1/7 9. 𝑥 = 0 Exercise B. 11. 𝑥 = 2l𝑛(5) l𝑛(3) ≈ 2.929 13. 𝑥 = 2ln(3)+ln(11) 3l𝑛(3) ≈ 1.394 Exercise C. 17. 𝑥 = 5 19. 𝑥 = 3 21. 𝑥 = 3 23. 𝑥 = 3 15. 𝑥 = ln(3) 3l 𝑛 3 +l𝑛(5) ≈ 0.223 More on Log and Exponential Equations

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4.6 more on log and exponential equations

  • 1. More on Log and Exponential Equations http://www.lahc.edu/math/precalculus/math_260a.html
  • 2. In this section we continue with log-equations and exp-equations. More on Log and Exponential Equations
  • 3. In this section we continue with log-equations and exp-equations. They may be grouped in two types. More on Log and Exponential Equations
  • 4. In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). More on Log and Exponential Equations
  • 5. In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations
  • 6. In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators
  • 7. In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators The Law of Uniqueness of Log and Exp Functions:
  • 8. In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators If logb(U) = logb(V) then U = V. The Law of Uniqueness of Log and Exp Functions
  • 9. In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators If logb(U) = logb(V) then U = V. If expb(U) = expb(V) or bU = bV then U = V. The Law of Uniqueness of Log and Exp Functions
  • 10. In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators Use this law to simplify log or exp equations when both sides can be consolidated into a common base. The Law of Uniqueness of Log and Exp Functions If logb(U) = logb(V) then U = V. If expb(U) = expb(V) or bU = bV then U = V.
  • 11. In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators Use this law to simplify log or exp equations when both sides can be consolidated into a common base. An example of numbers with common bases are {.., 1/8, ¼, ½, 1, 2, 4, 8,..}. The Law of Uniqueness of Log and Exp Functions If logb(U) = logb(V) then U = V. If expb(U) = expb(V) or bU = bV then U = V.
  • 12. In this section we continue with log-equations and exp-equations. They may be grouped in two types. * Equations that do not need calculators (Equations with related bases on both sides). * Numerical equations that require calculators. More on Log and Exponential Equations Equations That Do Not Need Calculators Use this law to simplify log or exp equations when both sides can be consolidated into a common base. An example of numbers with common bases are {.., 1/8, ¼, ½, 1, 2, 4, 8,..}. All these numbers are base 2 numbers, i.e. they are powers of 2. The Law of Uniqueness of Log and Exp Functions If logb(U) = logb(V) then U = V. If expb(U) = expb(V) or bU = bV then U = V.
  • 13. Example A: Solve 2*42x – 1 = 81 – 3x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first.
  • 14. Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first.
  • 15. Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first.
  • 16. Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv,
  • 17. Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x 2*24x – 2 = 23 – 9x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv,
  • 18. Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x 2*24x – 2 = 23 – 9x 21 + 4x – 2 = 23 – 9x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv,
  • 19. Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x 2*24x – 2 = 23 – 9x 21 + 4x – 2 = 23 – 9x drop the base 2 1 + 4x – 2 = 3 – 9x Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv, then drop the base b and solve U = V.
  • 20. Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x 2*24x – 2 = 23 – 9x 21 + 4x – 2 = 23 – 9x drop the base 2 1 + 4x – 2 = 3 – 9x 4x – 1 = 3 – 9x 13x = 4 Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv, then drop the base b and solve U = V.
  • 21. Example A: Solve 2*42x – 1 = 81 – 3x Since 4 = 22 and 8 = 23, put both sides of the equation in base 2 as 2*42x – 1 = 81 – 3x 2*(22)2x – 1 = (23)1 – 3x 2*24x – 2 = 23 – 9x 21 + 4x – 2 = 23 – 9x drop the base 2 1 + 4x – 2 = 3 – 9x 4x – 1 = 3 – 9x 13x = 4  x = 4/13 Equations That Do Not Need Calculators For exp-equations of this type, put all the bases into a common base first. Then consolidate the exponents on each side and put the equation into the form bu = bv, then drop the base b and solve U = V.
  • 22. For log-equations of this type, consolidate the logs on each side first. Equations That Do Not Need Calculators
  • 23. For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. Equations That Do Not Need Calculators
  • 24. For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, Equations That Do Not Need Calculators Example B: Solve log2(x – 1) + log2(x + 3) = 5
  • 25. Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V,
  • 26. Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
  • 27. Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
  • 28. Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 x2 + 2x – 3 = 32 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
  • 29. Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 x2 + 2x – 3 = 32 x2 + 2x – 35 = 0 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
  • 30. Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 x2 + 2x – 3 = 32 x2 + 2x – 35 = 0 (x + 7)(x – 5) = 0 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
  • 31. Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 x2 + 2x – 3 = 32 x2 + 2x – 35 = 0 (x + 7)(x – 5) = 0 x = -7, x = 5 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
  • 32. Example B: Solve log2(x – 1) + log2(x + 3) = 5 Combine the log using product rule: log2[(x – 1)(x + 3)] = 5 Write it in exp-form: (x – 1)(x + 3) = exp2(5) = 25 x2 + 2x – 3 = 32 x2 + 2x – 35 = 0 (x + 7)(x – 5) = 0 x = -7, x = 5 Equations That Do Not Need Calculators For log-equations of this type, consolidate the logs on each side first. There are two following possibilities. I. If the resulting equation is of the form logb(U) = V, drop the log by writing it in the exponential form U = eV.
  • 33. II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), Equations That Do Not Need Calculators
  • 34. II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
  • 35. Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
  • 36. Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) Combine the logs using quotient rule: log4[ ] = log4(5)(x + 2) (x – 1) II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
  • 37. Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) Combine the logs using quotient rule: log4[ ] = log4(5) Drop the logs on both sides to get (x + 2) (x – 1) (x + 2) (x – 1) = 5 II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
  • 38. Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) Combine the logs using quotient rule: log4[ ] = log4(5) Drop the logs on both sides to get x + 2 = 5(x – 1) (x + 2) (x – 1) (x + 2) (x – 1) = 5 II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
  • 39. Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) Combine the logs using quotient rule: log4[ ] = log4(5) Drop the logs on both sides to get x + 2 = 5(x – 1) x + 2 = 5x – 5 (x + 2) (x – 1) (x + 2) (x – 1) = 5 II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
  • 40. Example C: Solve log4(x + 2) – log4(x – 1) = log4 (5) Combine the logs using quotient rule: log4[ ] = log4(5) Drop the logs on both sides to get x + 2 = 5(x – 1) x + 2 = 5x – 5 7 = 4x 7/4 = x (x + 2) (x – 1) (x + 2) (x – 1) = 5 II. If after consolidating the logs, the resulting equation is of the form logb(U) = logb (V), just drop the log and solve the equation U = V. Equations That Do Not Need Calculators
  • 41. Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators.
  • 42. Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section.
  • 43. Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10.
  • 44. Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10. We do this by changing the calculation into base e with the following formula.
  • 45. Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10. We do this by changing the calculation into base e with the following formula. Given x and a base b, we've blog (x) = x,b
  • 46. Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10. We do this by changing the calculation into base e with the following formula. Given x and a base b, we've blog (x) = x, apply ln( ) to both sides: ln(blog (x)) = ln(x) b b
  • 47. Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10. We do this by changing the calculation into base e with the following formula. Given x and a base b, we've blog (x) = x, apply ln( ) to both sides: ln(blog (x)) = ln(x) use the power rule logb(x)ln(b) = ln(x) b b
  • 48. Numerical Equations That Require Calculators With numerical equations we seek both the exact and approximate answers with calculators. We have seen some examples in the last section. In this section, we tackle the equations that contain bases that are different from e and 10. We do this by changing the calculation into base e with the following formula. Given x and a base b, we've blog (x) = x, apply ln( ) to both sides: ln(blog (x)) = ln(x) use the power rule logb(x)ln(b) = ln(x) Therefore we've the Base-Change Formula logb(x) = b b ln(x) ln(b)
  • 49. Numerical Equations That Require Calculators To solve equations in bases different from e,
  • 50. Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first,
  • 51. Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
  • 52. Example D: (Base Change) Solve for x if 14 = 3·52x+1. Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
  • 53. Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
  • 54. Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
  • 55. Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Rewrite in log5( ): log5(14/3) = 2x+1 Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
  • 56. Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Rewrite in log5( ): log5(14/3) = 2x+1 Solve for x: log5(14/3) – 1 = 2x Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
  • 57. Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Rewrite in log5( ): log5(14/3) = 2x+1 Solve for x: log5(14/3) – 1 = 2x (log5(14/3) – 1) / 2 = x Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
  • 58. Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Rewrite in log5( ): log5(14/3) = 2x+1 Solve for x: log5(14/3) – 1 = 2x (log5(14/3) – 1) / 2 = x By base-change formula using ln: ( – 1) / 2 = xln(14/3) ln(5) Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
  • 59. Example D: (Base Change) Solve for x if 14 = 3·52x+1. Isolate the exp-part: 14/3 = 52x+1 14/3 = exp5(2x+1) Rewrite in log5( ): log5(14/3) = 2x+1 Solve for x: log5(14/3) – 1 = 2x (log5(14/3) – 1) / 2 = x By base-change formula using ln: ( – 1) / 2 = x  -0.0214ln(14/3) ln(5) Numerical Equations That Require Calculators To solve equations in bases different from e, we obtain the exact solution in the given base first, then use the change of bases formula to obtain the approximate solutions.
  • 60. Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Numerical Equations That Require Calculators
  • 61. Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: Numerical Equations That Require Calculators
  • 62. Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) Numerical Equations That Require Calculators
  • 63. Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) Numerical Equations That Require Calculators
  • 64. Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Numerical Equations That Require Calculators
  • 65. Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Solve for x: x – 2log4(7)x = log4(7) + 1 Numerical Equations That Require Calculators
  • 66. Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Solve for x: x – 2log4(7)x = log4(7) + 1 (1 – 2 log4(7))x = log4(7) + 1 Numerical Equations That Require Calculators
  • 67. Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Solve for x: x – 2log4(7)x = log4(7) + 1 (1 – 2 log4(7))x = log4(7) + 1 x = Numerical Equations That Require Calculators log4(7) + 1 1 – 2log4(7)
  • 68. Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Solve for x: x – 2log4(7)x = log4(7) + 1 (1 – 2 log4(7))x = log4(7) + 1 x = Change bases x = Numerical Equations That Require Calculators + 1 1 – ln(7) ln(4) 2ln(7) ln(4) log4(7) + 1 1 – 2log4(7)
  • 69. Example E: (Base Change) Solve for x if 4x–1 = 72x+1. Apply log4( ) to both sides: log4(4x–1) = log4(72x+1) x – 1 = log4(7)(2x + 1) x – 1 = 2log4(7)x + log4(7) Solve for x: x – 2log4(7)x = log4(7) + 1 (1 – 2 log4(7))x = log4(7) + 1 x = Change bases x = =  -1.33 Numerical Equations That Require Calculators + 1 1 – ln(7) ln(4) 2ln(7) ln(4) log4(7) + 1 1 – 2log4(7) ln(7) + ln(4) ln(4) – 2ln(7)
  • 70. More on Log and Exponential Equations
  • 71. More on Log and Exponential Equations
  • 72. (Answers to the odd problems) Exercise A. 1. 𝑥 = 2/5 3. 𝑥 = 2/5 5. 𝑥 = 3/10 7. 𝑥 = 1/7 9. 𝑥 = 0 Exercise B. 11. 𝑥 = 2l𝑛(5) l𝑛(3) ≈ 2.929 13. 𝑥 = 2ln(3)+ln(11) 3l𝑛(3) ≈ 1.394 Exercise C. 17. 𝑥 = 5 19. 𝑥 = 3 21. 𝑥 = 3 23. 𝑥 = 3 15. 𝑥 = ln(3) 3l 𝑛 3 +l𝑛(5) ≈ 0.223 More on Log and Exponential Equations