PROPERTIES OF LOGARITHMS

SOLVING LOGARITHMIC EQUATIONS

1. To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable.

Example 7: Solve for x in terms of b in the equation

displaymath85


Solution:

Step 1: The term tex2html_wrap_inline87 is valid when tex2html_wrap_inline89 , and the term tex2html_wrap_inline91 is valid when x>0. If we restrict the domain to the set of all real numbers x between 0 and tex2html_wrap_inline95 , every term in the equation is valid.
You can also graph the function

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using a positive value for b and note that the entire graph is located between the values x = 0 and tex2html_wrap_inline99 .
Note also that the value of the base b must be greater than zero.

Step 2: Simplify the original equation by gathering the logarithmic terms to the left side of the equal sign:

displaymath101

Step 3: Simplify the left side of the equation using Logarithmic Rule 2:

displaymath103

Step 4: Convert the above equation to an exponential equation with base b and exponent 3:

displaymath105

Step 5: Multiply both sides of the above equation by x:

displaymath107

Step 6: Add 3x to both sides of the above equation:

displaymath109

Step 7: Factor the left side of the above equation:

displaymath111

Step 8: Divide both sides of the above equation by tex2html_wrap_inline113 :

displaymath115

You have solved for x in terms of b.

Check: Let's check the answer by substituting

displaymath115

in the original equation. If, after the substitution, the left side of the original equations equals the right side of the original equation, you have found the right answer: Does

displaymath119

The term

displaymath121

can be simplified to

displaymath123

which is equal to the right side of the equation..
You have correctly worked the problem.

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

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Author: Nancy Marcus

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