1. Solving Log and Exponential Equations We have solved simple log and exponential equations by setting either the exponents equal to each other or the pieces we are taking the logs of equal to each other. Nate is god

2. Solving Log and Exponential Equations Example: log 2 (4) + log 2 (x) = log 2 (6) + log 2 (2) Condense each side to a single log log 2 (4x) = log 2 (6*2) log 2 (4x) = log 2 (12) Set the parenthesis equal to each other 4x = 12 4 4 x = 3

3. What happens if we can’t get a single log on each side? log 5 (3x + 1) = 2 We need a new method to solve this In order to solve any log function where we can’t get a single log on each side we need to use the following principle

4. log 5 (3x + 1) = 2 Remember that log functions and exponential functions are related to each other – you can change from one form to another. So when solving a log function with a log on only one side you will solve by doing the following:

5. log 5 (3x + 1) = 2 1) First you will get a single log all by itself on one side of the equation and put back any exponents 2) Then convert the log form into the exponential form 3) You will now be able to solve the equation

6. Examples: 4log 3 (x) = 28 3 28 = x 4 Convert to exponential form log 3 (x 4 ) = 28 Put the Exponent Back The calculator gives you scientific notation – that’s ok, just hit Ans^(1/4) to get the 4 th root. 2187 = x = x 4 (() 1/4

7. Examples: 1 / 3 log 2 x + 5 = 7 Get the log by itself - 5 -5 1 / 3 log 2 x = 2 Put the exponent back log 2 (x 1/3 ) = 2 Change to exponential form 2 2 = x 1/3 4 = x 1/3 (()3)3 )3)3 64 = x Raise each side to the 3 rd power (the opposite of the 1/3 rd power)

8. Solving ln equations: 16ln(x) = 30 ln(x 16 ) = 30 Put the exponent back log e (x 16 ) = 30 Change to log e form e 30 = x 16 Enter e^30 in your calculator, then raise that answer to the 1/16 th power to get the final answer Change to exponential form 6.521 = x

9. Try These: 6ln(4x) – 1 = 15 Put the exponent back Change to log e form ln(4x) 6 = 16 Enter e^16 in your calculator, divide by 4096, then raise that answer to the 1/6 th power to get a final answer. Change to exponential form e 16 = (4x) 6 Get the ln by itself +1 +1 6ln(4x) = 16 log e (4x) 6 = 16 e 16 = 4 6 x 6 e 16 = 4096x 6 3.046 = x

10. Solving exponential equations: 10 x + 5 = 60 - 5 - 5 Get the term with the exponent alone 10 x = 55 log 10 55 = x Use your calculator to get the answer log(55)/log(10) Change to log form 1.740 = x

11. Another Example: 10 -12x + 6 = 100 - 6 - 6 Get the term with the exponent alone 10 -12x = 94 log 10 94 = -12x Use your calculator to get the answer log(94)/log(10) Change to log form 1.973 = -12x -12 -.164 = x

12. Solving equations with e: 4e 2x = 5 Get the term with the exponent alone e 2x = 1.25 log e 1.25 = 2x Use your calculator to get the answer ln(1.25) Change to log form.2231 = 2x 4 4 Change to ln form ln1.25 = 2x 2 2.1116 = x

13. One More Example: 4 – 2e x = -23 Get the term with the exponent alone -2e x = -27 Use your calculator to get the answer ln(13.5) Change to log form log e (13.5) = x Change to ln form e x = 13.5 -4 - 4 -2 ln(13.5) = x 2.6 = x