SOLUTION: Write the expression as a single logarithm. Express powers as factors. (Simplify your answer) ln(x/(x-2))+ln((x+2)/x)-ln((x^2)-4)

Question 365204: Write the expression as a single logarithm. Express powers as factors. (Simplify your answer)
ln(x/(x-2))+ln((x+2)/x)-ln((x^2)-4)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
I assume the expression is:

These are not like terms because the arguments are different. But there are properties of logarithms which allow us to combine these logarithms into one:

With these properties all you need is the same base and coefficients of 1 and your logarithms meet both requirements. Using the first property on the first two logarithms (because of the "+" between them) we get:

The x's cancel:

leaving:

Now we can use the second property (because of the "-" between them):

Now we try to simplify the compllex fraction. We start by factoring the denominator:

Now we will multiply the numerator and denominator by (x-2):

The (x-2)'s in the numerator cancel:

leaving:

which simplifies to:

Now the (x+2)'s cancel:

leaving:

Since it says not to use powers, we will not change the denominator to . This may be an acceptable answer. However we can simplify a little further. We can rewrite the argument as follows:

and then use the thrid property of logarithms,
or

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