SOLUTION: Express as a single logarithm, and, if possible, simplify. 1/4 log[a] x + 8 log[a] y

SOLUTION: Express as a single logarithm, and, if possible, simplify. 1/4 log[a] x + 8 log[a] y - 2 log[a] x

Question 285093: Express as a single logarithm, and, if possible, simplify. 1/4 log[a] x + 8 log[a] y - 2 log[a] x
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

There are two ways to combine logarithmic terms:

By adding or subtracting them. But this requires that both the bases of the logarithms and their arguments be the same. For example:

But
cannot be added because the arguments are different. And
cannot be added because the bases are different.
Use the following properties of logarithms:
These properties require that the bases are the same and that the coefficients of the logarithms are 1's.

Your logarithms cannot be added or subtracted because the arguments are different. And we cannot use the properties, yet, because the coefficients are not 1's. But there is another property of logarithms, , which allows us to move a coefficient of a logarithm into its argument as an exponent. So by using this property on your logarithms we can make the coefficients 1's which, in turn, allows us to use the earlier properties to combine them! So we start by using this third property to move the coefficients into the arguments as exponents:

Now that the coefficients are 1's we can use the first two properties. Since the first two logarithms have a "+" between them we use the first property to combine them:

The remaining logarithms have a "-" between them so we use the second property to combine them:

We now have a single logarithm. In the argument, we can use the rule for exponents to divide the x's. The rule is to subtract the exponents adn :

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