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Solving Logarithmic Equations

Use technology to solve equations with logs

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Solving Logarithmic Equations

"I'm thinking of a number," you tell your best friend. "The number I'm thinking of satisfies the equation log10x2logx=3." What number are you thinking of?

Solving Logarithm Equations

A logarithmic equation has the variable within the log. To solve a logarithmic equation, you will need to use the inverse property, blogbx=x, to cancel out the log.

Let's solve the following logarithmic equations.

  1. log2(x+5)=9

There are two different ways to solve this equation. The first is to use the definition of a logarithm.

log2(x+5)29512507=9=x+5=x+5=x

The second way to solve this equation is to put everything into the exponent of a 2, and then use the inverse property.

2log2(x+5)x+5x=29=512=507

Make sure to check your answers for logarithmic equations. There can be times when you get an extraneous solution. log2(507+5)=9log2512=9

  1. 3ln(x)5=10

First, add 5 to both sides and then divide by 3 to isolate the natural log.

3ln(x)53ln(x)ln(x)=10=15=5

Recall that the inverse of the natural log is the natural number. Therefore, everything needs to be put into the exponent of e in order to get rid of the log.

eln(x)xx=e5=e5=e5148.41

Checking the answer, we have 3ln((e5))5=103lne55=10355=10

  1. log5x+log(x1)=2

Condense the left-hand side using the Product Property.

log5x+log(x1)=2log[5x(x1)]=2log(5x25x)=2

Now, put everything in the exponent of 10 and solve for x.

10log(5x25x)5x25xx2x20(x5)(x+4)x=102=100=0=0=5,4

Now, check both answers.

log5(5)+log(51)log25+log4log100=2log5(4)+log((4)1)=2=2 log(20)+log(5)=2=2

-4 is an extraneous solution. In the step log(20)+log(5)=2, we cannot take the log of a negative number, therefore -4 is not a solution. 5 is the only solution.

Examples

Example 1

Earlier, you were asked to determine what number you are thinking of if the number satisfies the equation log10x2logx=3.

We can rewrite log10x2logx=3 as log10x2x=3 and solve for x.

log10x2x=3log10x=310log10x=10310x=1000x=100

Therefore, the number you are thinking of is 100.

Example 2

Solve: 9+2log3x=23.

Isolate the log and put everything in the exponent of 3.

9+2log3x2log3xlog3xx=23=14=7=37=2187

Example 3

Solve: ln(x1)ln(x+1)=8.

Condense the left-hand side using the Quotient Rule and put everything in the exponent of e.

ln(x1)ln(x+1)ln(x1x+1)x1x+1x1x1xxln8x(1ln8)x=8=8=ln8=(x+1)ln8=xln8+ln8=1+ln8=1+ln8=1+ln81ln82.85

Checking our answer, we get ln(2.851)ln(2.85+1)=8, which does not work because the first natural log is of a negative number. Therefore, there is no solution for this equation.

Example 4

Solve: 12log5(2x+5)=5.

Multiply both sides by 2 and put everything in the exponent of a 5.

12log5(2x+5)log5(2x+5)2x+52xx=2=4=625=620=310

Review

Use properties of logarithms and a calculator to solve the following equations for x. Round answers to three decimal places and check for extraneous solutions.

  1. log2x=15
  2. log12x=2.5
  3. log9(x5)=2
  4. log7(2x+3)=3
  5. 8ln(3x)=5
  6. 4log33xlog3x=5
  7. log(x+5)+logx=log14
  8. 2lnxlnx=0
  9. 3log3(x5)=3
  10. 23log3x=2
  11. 5logx23log1x=log8
  12. 2lnxe+2lnx=10
  13. 2log6x+1=log6(5x+4)
  14. 2log12x+2=log12(x+10)
  15. 3log23xlog2327=log238

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 8.11. 

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