SOLUTION: Compress the expression {{{2(ln(x-3)+ln(x))-ln(x^(2)-9)}}} into one simplified logarithm.

Question 258258: Compress the expression into one simplified logarithm.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

To combine logarithms we use two of the properties of logarithms:

These properties require that the coefficients of the logarithms are 1's. For logarithms which have other coefficients, we have a third property which allows us to move a coefficient of a logarithm into the argument as an exponent:
Inside the parentheses we find . Since this is an addition, we can use the first property to combine these:

Next, we can use the third property to move the 2 from in front into the argument of the logarithm:

And finally, since this is a subtraction, we can use the second property to combine the remaining logarithms:

We have now condensed the expression into a single logarithm. This may be an acceptable answer. But we can simplify the argument of the logarithm. The fraction will reduce. After we factor the denominator we get:

We can see that the x-3 in the denominator will cancel with one of the two (x-3)'s (it is squared, after all) leaving:

Last of all we can multiply out the numerator:

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