Expanding Logarithms - Problem 3

Expanding logarithms. Whenever we expand a logarithm what we want to do is to take a complicated log like this one here and rewrite it as the sum of a series of different logs, in more simple form. For this one what I’m going to do is walk you through expanding it by going through some shortcuts.

So what we have is a huge inside to a negative power. A negative power is associated with everything so we can always bring it out in front to start. So this becomes negative 3, log base 4 64 over 3 x to the 4th y². For this one what I’m actually going to do is skip some steps.

We know that -3 is associated with everything so I’m going to put that on the outside with a parenthesis and fill in the gaps. I know that anything in the numerator of this fraction is going to end up positive, so what I know then is have log base 4 64 is going to be a positive coefficient. I also know that everything in the denominator because being divided is going to turn out to be negative. So this then turns into minus log base 4 cube root of x to the 4th minus log base 4 y².

Looking at this now, evaluating it up a little bit as we go, cube root of x to the 4th, remember power over root so this just turns into x to the 4/3. You can bring the 4/3 down to the front so this becomes minus 4/3 log base 4 of x, y² we can bring the 2 down in front. Minus 2 log 4 of y and log base 4 of 64, what power of 4 gives us 64? That’s going to be 3, and we still have this -3 out in front.

Using some basic principles, if a term in the numerator is going to turn up positive, the term is in the denominator it’s going to end up negative and every time you have a power you can bring it up in front. We have evaluated this out as much as we can and expanded it out. You could distribute the -3 through if you wanted to, I’m going to leave it out just to make life a little bit easier but if you’re needed to just distribute it through as you would normally.