Solving Basic Logarithmic Equations

To solve a logarithmic equation:

1. Get the logarithm by itself on one side of the equation.

2. Determine the base of the logarithm.

3. Raise both sides of the equation to be a power of that base.

4. Simplify and solve.

Examples:

1) 

The logarithm is already by itself. The base of the log is 10, so we must raise both sides of the equation to be powers of 10:



On the left hand side, the 10 and log cancel, leaving just 2x.



2x = 10,000

x = 5,000

We can check this answer by substituting it back in for x.


2) 

First, we'll move the 4 to the other side to get the natural log by itself:



The base of the log is e, so we must raise both sides of the equation to be powers of e:



On the left hand side, the e and ln cancel, leaving just 3x.





This is the exact answer, but we can find an estimation using our calculator:



We can check this answer by substituting it back in for x.


3) 

First we must get the log by itself by moving the 2 and the 5:





The base of the log is 3, so we must raise both sides of the equation to be powers of 3:



On the left hand side, the 3 and cancel, leaving just x - 1.



x - 1 = 9

x = 10


Practice: Solve each logarithmic equation. If necessary, round the answer to the nearest thousandth.

1)

2)

3)

4)

5)

Answers: 1) 25,000 2) 24.086 3) ±1024 4) 4202.5 5) 3.794


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Math
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