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Solving Log and Exponential Equations

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. Solving Log and Exponential Equations. Example:.

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Solving Log and Exponential Equations

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  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: log2(4) + log2(x) = log2(6) + log2(2) Condense each side to a single log log2(4x) = log2(6*2) log2(4x) = log2(12) Set the parenthesis equal to each other 4x = 12 x = 3 4 4

  3. What happens if we can’t get a single log on each side? log5(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. log5(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. log5(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: 4log3(x) = 28 Put the Exponent Back log3(x4) = 28 Convert to exponential form 328 = x4 ( )1/4 = x4 ( )1/4 The calculator gives you scientific notation – that’s ok, just hit Ans^(1/4) to get the 4th root. 2187 = x

  7. Examples: 1/3log2x + 5 = 7 Get the log by itself - 5 -5 1/3log2x = 2 Put the exponent back log2(x1/3) = 2 Change to exponential form 22 = x1/3 ( 4 = x1/3 )3 ( )3 Raise each side to the 3rd power (the opposite of the 1/3rd power) 64 = x

  8. Solving ln equations: 16ln(x) = 30 Put the exponent back ln(x16) = 30 Change to loge form loge(x16) = 30 Change to exponential form e30 = x16 Enter e^30 in your calculator, then raise that answer to the 1/16th power to get the final answer 6.521 = x

  9. Try These: 6ln(4x) – 1 = 15 Get the ln by itself +1 +1 6ln(4x) = 16 Put the exponent back ln(4x)6 = 16 Change to loge form loge(4x)6 = 16 Change to exponential form e16 = (4x)6 Enter e^16 in your calculator, divide by 4096, then raise that answer to the 1/6th power to get a final answer. e16 = 46x6 e16 = 4096x6 3.046 = x

  10. Solving exponential equations: 10x + 5 = 60 Get the term with the exponent alone - 5 - 5 10x = 55 Change to log form log1055 = x Use your calculator to get the answer log(55)/log(10) 1.740 = x

  11. Another Example: 10-12x + 6 = 100 Get the term with the exponent alone - 6 - 6 10-12x = 94 Change to log form log1094 = -12x Use your calculator to get the answer log(94)/log(10) 1.973 = -12x -12 -12 - .164 = x

  12. Solving equations with e: 4e2x = 5 Get the term with the exponent alone 4 4 e2x = 1.25 Change to log form loge1.25 = 2x Change to ln form ln1.25 = 2x Use your calculator to get the answer ln(1.25) .2231 = 2x 2 2 .1116 = x

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

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