SOLUTION: Use the logarithm properties to condense: 1/3[2 ln(x+5) - lnx

SOLUTION: Use the logarithm properties to condense: 1/3[2 ln(x+5) - lnx - ln(x^2-4)]

Question 371352: Use the logarithm properties to condense:
1/3[2 ln(x+5) - lnx - ln(x^2-4)]
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

The three logarithms in the parentheses are not like terms so we cannot subtract them. However there is a property of logarithms, , which allows us to combine two logarithms into one if all of the following are true:

There is a mimus between them.
The bases of the logarithms are the same.
The coefficient of each logarithm is a 1.

Your logarithms meet the first two requirements. But the first logarithm has a coefficient of 2. Fortunately there is another property of logarithms, , which allows us to "move" a coefficient into the argument as an exponent. So we start by using this property on the first logarithm:

Now we can use the first property on the first two logarithms:

Next we can use the first property again to combine the remaining logarithms:

which simplifies to:

This may be condensed enough. But we can use the second property again to move the coefficient of 1/3:

This may be the desired answer. Alternatively, since 1/3 as an exponent means "cube root", we could rewrite this as:

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