5-4 Exponential and Logarithmic Equations.ppt
- 1. 5-4 Exponential & Logarithmic Equations Strategies and Practice
- 2. Objectives – Use like bases to solve exponential equations. – Use logarithms to solve exponential equations. – Use the definition of a logarithm to solve logarithmic equations. – Use the one-to-one property of logarithms to solve logarithmic equations.
- 3. Use like bases to solve exponential equations • Equal bases must have equal exponents EX: Given 3x-1 = 32x + 1 then x-1 = 2x+1 x = -2 If possible, rewrite to make bases equal EX: Given 2-x = 4x+1 rewrite 4 as 22 2-x = 22x+2 then –x=2x+2 x=-2/3 Note: Isolate function if needed 3(2x)=48 2x =16
- 4. You try… 1. 4x = 83 2. 5x-2 = 25x 3. 6(3x+1) = 54 4. e–x2 = e-3x - 4
- 5. Exponentials of Unequal Bases • Use logarithm (inverse function) of same base on both sides of equation EX: Solve: ex = 72 lnex = ln72 xlne = ln72 x = ln72 (calc ready form) x ~ 4.277 EX: Solve: 7x-1 = 12 log77x-1 = log712 (x-1)log77 = log712 x-1 = log712 x = 1+log712 x ~ 1.277
- 6. You try… 1. Solve 3(2x) = 42 2. Solve 32t-5 = 15 3. Solve e2x = 5 4. Solve ex + 5 = 60
- 7. Solving Logarithmic Equations • Convert to exponential (inverse) form EX: Solve: lnx = -1/2 elnx = e-1/2 x = e-1/2 x .607 EX: Solve: 2log53x = 4 log53x = 2 (get the log by itself) 52 = 3x 25/3 = x Use Properties of Logs to condense EX: Solve: log4x + log4(x-1) = ½ log4(x2-x)= ½ 41/2 = x2 – x 0 = x2-x-2 (x-2)(x+1) x=2
- 8. You try… 1. Solve lnx = -7 2. Solve 2log5 3x = 4 3. Solve . lnx+ln(x-3) = 1 4. Solve . 5 + 2ln x = 4
- 9. Double-Sided Log Equations • Equate powers (domain solutions only) EX: Solve: log5(5x-1) = log5(x+7) 5x – 1 = x + 7 x = 2 EX: Solve: ln(x-2) + ln(2x-3) = 2lnx Use a property: ln(x-2)(2x-3) = lnx2 2x2 – 7x + 6 = x2 x2-7x+6=0 x = 6 & 1
- 10. You try… 1. Solve ln3x2 = lnx 2. Solve log6(3x + 14) – log6 5 = log6 2x 3. Solve log2x+log2(x+5) = log2(x+4)
- 11. SUMMARY • Equal bases Equal exponents • Unequal bases Apply log of given base • Single side logs Convert to exp form • Double-sided logs Equate powers Note: Any solutions that result in a log(neg) cannot be used!