PROPERTIES OF LOGARITHMS

SOLVING LOGARITHMIC EQUATIONS

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

Problem 4: Solve for x in the equation

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Answer: x = 2

Solution:

As you know by now, we can only take the logarithm of a positive number. Therefore, we will have to restrict the domain (values of x) so that the original equation is valid.
Step 1: The term tex2html_wrap_inline67 is valid when x is greater than zero, and the term tex2html_wrap_inline69 is valid when x is greater than 1. If we restrict the domain to the set of all real numbers that are greater than 1, both terms will be valid.
Step 2: Simplify the left side of the original equation using Logarithmic Rule 2:

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Step 3: Convert the logarithmic term to an exponential term with base 4 and exponent tex2html_wrap_inline73 :

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Step 4: Simplify the left side of the above equation:

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Step 5: Multiply both sides of the above equation by (x - 1):

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Step 6: Expand the left side of the above equation:

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Step 7: Subtract x from both sides of the above equation and add 2 to both sides of the above equation:

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Check: Let's substitute the value x = 2 in the original equation and determine whether the left side of that equation equals the right side of that equation after the substitution

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Since the value of the left side of the equation is tex2html_wrap_inline73 after you substitute the value 2 for x, and the right side of the equal is tex2html_wrap_inline73 , you have proved your answer.

If you would like to work on another problem, click on Problem.

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Author: Nancy Marcus

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