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 4:
Note:
The above equation is valid only if all of the terms are valid. The first
term is valid if 
the second term is
valid if 
the third term is valid if 
, and the fourth term is valid if 
Therefore, the equation is valid when all four
of these conditions are met, or when 
The domain is the set
of real numbers greater than 
.
Simplify both sides of the equation using the rules of logarithms.
Recall that when 
a = b. Therefore, if
Solve for x.
The exact answers are 
and the approximate answers are
1.69771560359 and -37.6977156036. Neither of these answers is located in
the restricted domain. Therefore, there are no real solutions to this
problem.
If you did not make this observation, you can catch it in the check.
Check the answer 
by substituting 1.69771560359 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.
Check the answer 
by substituting -37.6977156036 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.
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 never crosses the x-axis. This means that
there are no real solutions.
If you would like to test yourself by working some problems similar to this example, click on problem.
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