Using the properties of logarithms: multiple steps

we're asked to simplify log base 5 of 25 to the X power over Y so we can use some logarithm properties and I do agree that this does require some simplification over here that this having this right over here inside of the logarithm is not a pleasant thing to look at so the first thing that we realize and this is the logarithm one of our logarithm properties this logarithm for a given base so let's say that the base is X of a over B that is equal to log base X of a - log base X of B and here we have 25 to the x over Y so we can simplify so let me write this down this in blue log base 5 of 25 to the x over Y using this property means that it's the same thing as log base 5 of 25 to the X power minus log base 5 of Y now this looks like we can do a little bit of simplifying and it seems like the relevant logarithm property here is if I have log base X of a to the B power that's the same thing as B times log base X of a that this exponent over here can go be moved out front which is what we did right over there so this part right over here can be re-written as x times the logarithm base 5 of 25 and then of course we have minus log base 5 of y and this is useful because log base 5 of 25 is actually fairly easy to think about this part right here is asking us what power do I have to raise 5 to to get to 25 so we have to raise 5 to the second power to get to 25 so this simplifies to 2 so then we are left with this is equal to 2 and I'll write it in front of the X now 2 times X 2 times X minus minus log base 5 of Y and we're done