SOLUTION: I am going to try to write this question so that it is easy to understand...I hope. I am writing it out, as I am not real sure how to write it using a formula that may be available

Question 275924: I am going to try to write this question so that it is easy to understand...I hope. I am writing it out, as I am not real sure how to write it using a formula that may be available through this site. I have tried several times and cannot get the correct answer. I don't understand how to get the final answer. Such as 1/x or 1/x^2 and so on.
It says: Use the properties of the logarithms to write the expression as one logarithm. I can't find what property has anything to do with the square root symbol. That is the part that mainly confuses me.
4 log base 4 2(SQRT(x) - 3 log base of 4 x. It is a square root with a 2 in the index and an x underneath. Thanks in advance if anyone can explain this.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Problems like these are meant to help you learn how to use the properties of logarithms properly and effectively. These two terms are not like terms (the bases of the logarithms are the same but the arguments are different) so we cannot use regular subtraction to combine them into one. But there are some properties of logarithms which allow us to combine logarithms with the same base:

Both of these properties require that the two logarithms must have 1's in front. So what do we do if there are other numbers in front (like the 4 and the 3 in front of your logarithms)? Fortunately there is a third property, , which allows us to move a number in front of a logarithm into its argument as the exponent.

So we will start with the third property to move the 4 and the 3 "out of the way":

Simplifying the two arguments we get:

Now we can use the second property (because of the "-" between them) to combine the two logarithms into one:

Simplifying the argument we get:

This may be an acceptable answer.

But seeing a 4 in a base 4 logarithm tells me we can simplify this a bit more:

(third property "in reverse")
(first property "in reverse")
(since )
(Distributive Property)

Both and use a single logarithm in expressing your original expression in a simpler form. Which will be preferred by your teacher I cannot say.
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