SOLUTION: write 1/2[lnx+ln(x+2)]-2ln(x-5)as a single logarithmic expression

Question 633547: write 1/2[lnx+ln(x+2)]-2ln(x-5)as a single logarithmic expression
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Note: Algebra.com's formula drawing software will not allow brackets. So I changed your brackets to a set of parentheses:

We have three logarithms. We want only one. Somehow we must find a way to condense/combine the three into one.

There are two ways to combine logarithmic terms:

Add or subtract them. As usual, however, the terms must be like terms. Like logarithmic terms have the same bases and the same arguments.
Use either of the following properties:
These properties require that the bases be the same and the coefficients be 1's.

Your logarithms all have the same base (the base of ln is "e"). But the arguments are different so we cannot add or subtract them. The coefficients are not 1's either so we cannot use the properties... at least not yet. Fortunately there is another property of logarithms, , which allows us to "move" a coefficient into the argument as its exponent. With this third property we can get the coefficients of 1's we need for the other properties.

The first two logs, in the parentheses, already have the same base and coefficients of 1. So we can use the properties on them right now. We will use the first property be cause its logs, like ours, have a "+" between them.

which simplifies to:

Neither of these two logs have coefficients of 1. So we will need to use the third property to move them into the arguments as exponents:

Since an exponent of 1/2 means the same thing as square root, I'm going to rewrite the first argument as a square root:

Now that the coefficients are 1's we can use the other properties. We will use the second one because its logs, like ours, have a "-" between them:

This may well be the desired answer to your problem. Or we could multiply out the denominator:

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