SOLVING LOGARITHMIC EQUATIONS
SOLVING LOGARITHMIC EQUATIONS
Note:
If you would like an in-depth review of logarithms, the rules of logarithms, logarithmic functions and logarithmic equations, click on logarithmic function.
Solve for x in the following equation.
Example 1:
The above equation is valid only if all of the terms are valid. The first
term is valid if x - 2 > 0 or x > 2 ,
the second term is valid if 2x - 3 > 0 or
and the third term is valid if x>0.
Therefore, the
equation is valid when all three of these conditions are met, or when x>2 .
The domain is the set of real numbers greater than 2.
Simplify both sides of the equation using the rules of logarithms.
Recall that if 
then
a=b . Therefore, if
Solve for x.
The exact answer is 
Check the answer x=6 by substituting 6 in the original equation for x.
If the left side of the equation equals the right side of the equation after
the substitution, you have found the correct answer.
Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 6 for x, then x=6 is a solution.
You can also check your answer by graphing 
(formed by subtracting the right side of the original
equation from the left side). Look to see where the graph crosses the
x-axis; that will be the real solution. Note that the graph crosses the
x-axis at 6. This means that 6 is the real solution.
If you would like to work another example, click on example.
If you would like to test yourself by working some problems similar to this
example, click on problem.
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contents.
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