PROPERTIES OF LOGARITHMS

SOLVING LOGARITHMIC EQUATIONS

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

Problem 3: Solve for x in the equation

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Answer: x=e-1 is the exact answer and x=1.71828182846 is an approximate answer.

Solution:

Step 1: Simplify the left side of the equation using Logarithmic Rule 3:

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Step 2: tex2html_wrap_inline43 can be simplified to tex2html_wrap_inline45 only if we restrict the domain (values of x) so that the quantity x+1>0 or x>-1. Recall that we have to make this restriction on the domain because you can only take the logarithm of a positive number.
Step 3: Isolate the Ln term by dividing both sides by 2:

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Step 4: Convert the logarithmic equation to an exponential equation with base e and exponent 1:

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is the exact answer and

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is an approximate answer.

Check: Let's substitute the approximate value x=1.71828182846 in the original equation 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 it is only a close approximation. Therefore, the left and right sides of the equation will most likely be very close but may not equal; it depends on the number of decimals you kept in the rounding process.

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Since the value of the left side of the equation is 2 when you substitute the value of x, and the right side of the equations is 2, 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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