# SOLVING LOGARITHMIC EQUATIONS

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

Example 8: If , and a and b are positive, show that

no matter what positive value for the base is used for the logarithms (but it is understood that the same base is used throughout).

Solution:

Step 1: Rewrite the above equation using Logarithmic Rules 1 and 3:

All we did to the quantity with the brackets on the left side of the equation is raise it to a net power of 1. Remember that . All we did to the right side was simplify it using Logarithm Rules 1 and 3.

Step 2: Simplify the left side of the above equation:

Step 3: Since , the above term can be simplified to

Step 4: Simplify the above term:

or .

Step 5: The right side of the original equation,

can be simplified using Logarithmic Rules 1 and 3:

You have just shown that the left side of the original equation can be simplified to the right side of the original equation.

If you would like to review another example, click on Example.

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