Evaluating a Logarithmic Expression in terms of Known Quantities - Problem 2

Often times we will be given information about certain logs and we are supposed to use that information to simplify another statement. What we want to do when we are dealing with that is look to see how the numbers in that statement relate to the numbers that we are given.

So what I see in this case is I have our log base 5 of 4 over 9 I have the log of a quotient so therefore I know that I can break this up as subtraction, so split that 4 and that 9 component up into their own longs. So let's go ahead and of that so we end up with log base 5 of 4 minus log base 5 of 9.

The next thing we need to do is somehow try to express log base 5 of 4 in log base 5 of 9 using log base 5 of 2 or 3. So we have to go back to another property of logarithms which is if we have a power on our log, we can bring that to the outside, so I think about 4. 4 is actually 2 and 2² and 2 we have information about. So what we can do here is just rewrite this as log base 5 of 2² minus log base 5 of 3².

Those exponents we can then bring to the front using our properties of logs, so we end up with 2 log base 5 of 2 minus 2 log base 5 of 3. Now we have information about log base 5 of 2 and log base 5 of 3 already up here, so just simplifying that by plugging those in, this ends up being a 'j', this ends up being a 'k' and we end up with 2j minus 2k.

So using our properties of logs and just what we know about numbers to be able to express four-ninths as powers of 2 and powers of 3.