Expanding logarithms. So what we do when we expand logarithms is using the properties of logs that we know, we can try to rewrite a log in terms of as many simple logs as possible.
When we do that, what we have here is a log with three components inside of it. We have a number, and an x and a z. There’s numbers of ways of tackling this. What I sort of do is I first say, okay, what we have is a quotient. We have a numerator and a denominator, whenever we’re dividing inside our logarithms that comes out subtraction, so we can rewrite this as log base 5 of 25, x² minus log base 5 square root of z.
What we can do now is looking at this 25x². Common mistake that people want to do is to take this 2 out right now, because of the power rule we can take the exponent and bring it outside, but the thing you have to remember the 2 is not associated with this 25, so what we’re really doing is we’re multiplying the 25 and the x². We need to split this up using the product rule. Splitting this up, log base 5 of 25 plus log base 5, x², so we’re multiplying we could split up as addition and we still have this z term over here, log base 5 root z.
Now we still can go further. Log base 5 of 25, 5 to what power is 25, we know that this is 2. By the power rule we have x to a power that can come out in front, so 2 log 5 of x. And lastly with the z. We have the square root of z but if you remember square root is just to the power of ½, so we can take our ½ out in front, ½ log base 5 of z.
Using the various rules that we know of logarithms, namely the product rule, the quotient rule and the power rule, we were able to rewrite this very complicated log in terms of a series of much more simple logs. And just what you’ll end up noticing the much more you do this, is the terms that were in the numerator are going to end up being positive, so here we had the 25 and the x², the 2 turned positive, the x term turned positive and we had the z in the denominator, that was negative.
So it’s always a good way to check your work, check your signs, make sure they match up, numerator and denominator, positive, negative.