# SOLVING LOGARITHMIC EQUATIONS

1. To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable.

Example 9: Solve for x (assuming a>b>0) in the equation

Solution:

Step 1: Take the natural log of both sides of the above equation:

Step 2: Simplify the left side of the above equation using Logarithmic Rule 3 and simplify the right side of the equation using Logarithmic Rule 1:

Step 3: Simplify the left side of the equation by factoring and simplify the right side of the equation using Logarithmic rules:

Step 4: Simplify the left side of the above equation using Logarithmic Rule 3:

Step 5: Simplify the left side of the above equation:

or

Step 6: Now you can see why the initial restrictions on the values of a and b were such that a and b were both positive and a is larger than b.
Step 7: Gather all the x terms to the left side of the equation and all the non-x terms to the right sides of the equation:

which yields

or

Check: Let's check our answer using a=9 and b=5. You can check your answer with any positive values of a and b as long as the value of a is greater than the value of b (initial restriction).

If a = 9 and b = 5, then

When the values for a, b, and x are substituted in the left side of the initial equation, the value is as follows:

When the values for a, b, and x are substituted in the right side of the initial equation, the value is as follows:

The answer checks for these values of a and b.

If you would like to work on problems, click on Problems.

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