PROPERTIES OF LOGARITHMS
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 nonx terms to the right sides of the equation:
which yields
or
Answer:
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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Author: Nancy
Marcus
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