PROPERTIES OF LOGARITHMS

SOLVING LOGARITHMIC EQUATIONS

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

Problem 1: Solve for x in the equation

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Answer: tex2html_wrap_inline45 is the exact answer and x=104.142857143 is an approximate answer.

Solution:

Step 1: Since you cannot take the log of a negative number, we have to restrict the domain so that 7x >0 or x > 0.
Step 2: Isolate the Log term in the original equation by subtracting 4 from each side of the equation:

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Step 3: Convert the above logarithmic equation to an exponential equation with base 3 and exponent 6:

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Step 4: Divide both sides of the above equation by 7:

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is the exact answer and tex2html_wrap_inline59 is an approximate answer.

Check: Let's substitute the approximate value in the answer and determine whether the left side of the equation equals the right side of the equation after the substitution. Remember we rounded the number and the answer is only a close approximation, so the left and right side of the equation will most likely be very close but not equal; it depends on the number of decimals were rounded in your answer

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or

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Since the value of the left side of the equation is very close to 10 when you substitute the value of x, and the right side of the equal is 10, you have proved your answer. It won't check exactly because we rounded the value of x.

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

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Author: Nancy Marcus

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