Question 718443: How do you write 5log[base2]x+(1/2)log[base2]y as a single logarithm
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
We have two logs which we want to write as one log. There are two ways to combine two logarithmic terms into one:- If they are like terms we can add or subtract them (depending on whether there's a "+" or "-" between them. Like logarithmic terms have the same bases and the same argument.
- Use one of the following properties of logarithms:These properties require logs with the same base and coefficients of 1.
Our logs have the same bases, 2. But the arguments, x and y, are different. So we cannot add them together. Also our logs have coefficients, 5 and 1/2, which are not 1's. So we cannot use the properties... yet. Fortunately there is another property of logs,  , which will help. The right side shows a log with a coefficient which is not a 1. The left side shows us how to rewrite the right side with a coefficient of 1: Just move the coefficient into the argument as its exponent!
So we start by using the property which lets us move the coefficients:
Since an exponent of 1/2 means square root, I'm going to rewrite it as a square root:
Now we will use  (because its logs, like ours, have a "+" between them) to combine the logs:
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