PROPERTIES OF LOGARITHMS

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:

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

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