![A Trickier Example
Write as one log on one log(x + 21) + log(x) = 2
side log [(x+21)x] = 2 that when there is no base written on a lo
Remember
Use the definition of x2+21x = 102
logarithms to write in
exponential form x2+21x – 100 = 0
(x+25)(x-4) = 0
Solve for x
x = 4, -25
BUT we CANNOT take the log of a
negative number, so we will have
to throw out x = -25 as one of our
solutions](https://image.slidesharecdn.com/logsummaryequations-120430162412-phpapp02/85/Log-summary-equations-11-320.jpg)
Log summary & equations
- 2. Learning Intention Recap logarithmic expressions, rules & equations Success criteria: you will be able to Express log statements in exponential form Apply log rules Solve log equations
- 3. Back to Basics A log is just the inverse of an exponential! y = bx is equivalent to logb(y) = x (means the exact same thing as) Image source: www.purplemath.com
- 4. The Log Switcheroo Write the following exponential expressions in log form: 63 = 216 log6(216) = 3 45 = 1024 log4(1024) = 5 Write the following logarithmic expressions in exponential form: log2(8) = 3 23 = 8 52 = 25 log5(25) = 2 641/3 = 4 log64(4) = 1/3
- 5. SIMPLE (base 10) EXAMPLES Exponential Number Logarithm Expression 1000 103 3 100 102 2 10 101 1 1 100 0 1/10 = 0.1 10-1 -1 1/100 = 0.01 10-2 -2 1/1000 = 0.001 10-3 -3
- 6. Some Things To Remember b1 = b , so logb(b) = 1, for any base b b0 = 1 , so logb(1) = 0 logb(a) is undefined if a is negative logb(0) is undefined logb(bn) = n
- 7. Calculations with Logs Because logarithms are exponents, mathematical operations involving them follow the same rules as those for exponents: 1) Multiplication inside the log can be turned into addition outside the log, and vice versa. logb(mn) = logb(m) + logb(n) 2) Division inside the log can be turned into subtraction outside the log, and vice versa. logb(m/n) = logb(m) – logb(n) 3) An exponent on everything inside a log can be moved out front as a multiplier, and vice versa. logb(mn) = n · logb(m)
- 8. Log Rule Practice Expand log3(2x) log3(2x) = log3(2) + log3(x) Expand log4( 16/x ) log4( 16/x ) = log4(16) – log4(x) log4(16) = 2 so log4( 16/x ) = 2 – log4(x) Expand log5(x3) log5(x3) = 3 · log5(x) = 3log5(x)
- 9. Change of Base eg Evaluate log3(6) log3(6) = log(6) = 1.63092975... log(3)
- 10. Solving a Log Equation Step 1: Write as one log on one side Step 2: Use the definition of logarithms to write in exponential form (or vice versa) Step 3: Solve for x eg log5(x+2) = 3 53 = x + 2 125 = x + 2 x = 123
- 11. A Trickier Example Write as one log on one log(x + 21) + log(x) = 2 side log [(x+21)x] = 2 that when there is no base written on a lo Remember Use the definition of x2+21x = 102 logarithms to write in exponential form x2+21x – 100 = 0 (x+25)(x-4) = 0 Solve for x x = 4, -25 BUT we CANNOT take the log of a negative number, so we will have to throw out x = -25 as one of our solutions
- 12. Exam Question Write as one log on one H(t) = 3 + (1.24)t side When does H = 7? Use the definition of 7 = 3 + (1.24)t logarithms to write in exponential form (or 4 = (1.24)t vice versa) log (4) = log (1.24t) Solve for x log (4) = t x log (1.24) t= log (4) = 6.44 years log(1.24)
- 13. Try the Following: Write as one log on one log(2x – 4) = 3 side log2(x) + log2 (x – 6) = 4 Use the definition of logarithms to write in log4(x + 4) – log4(x – 1) = 2 exponential form Solve for x