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.
Problem 3:
Answer:
The exact answer is 
Solution:
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 x > 1.5. The domain is the set of real
numbers greater than 1.5.
Simplify both sides of the equation using the rules of logarithms.
Recall that if 
then a = b. Therefore, if
The exact answer is
Check the answer x = 14.5 by substituting 14.5 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 14.5 for x, then x = 14.5 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 14.5. This means that 14.5 is the real solution.
If you would like to review the solution to problem 8.3d, click on solution.
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