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We are asked to simplify a log base 5 of 25 to the "x" power over "y"

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So I will give you some logarithm properties and I do agree that this does require some simplification

3
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over here

4
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That this having this right over here inside of the logarithm is not a pleasant way to look at it

5
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So the first thing that we realize is that this is one of our logarithm properties

6
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So logarithm for a given base so lets say that the base is "x" of a/b

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That is equal to log base "x" of a minus log base "x" of b

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Now here we have 25 to the "x" over y

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So we can simplify

10
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Log base 5 twenty five to the "x" over "y" to this property means that its the same thing

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as log base 5 to the twenty five to the "x" power minus log base 5 of "y"

12
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Now this looks like we can do a little bit of simplifying

13
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It seems like the relevant logarithm property here is if I have log base "x" of a to the "p" power

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That's the same thing as b times log base "x" of a

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This exponent over here can go outfront, which is what we did right over there

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So this part right over here can be written as "x" times the logarithm of base 5

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of 25 and then of course we have minus log base 5 of "y"

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And this is useful beacuse log base 5 of 25 is actually very easy to think about

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This part right here is asking us what power

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do I have to raise 5 to to get to 25

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So we have to raise 5 to the 2nd power to get 25.

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So we are left with, this is equal to 2 times "x" minus log base 5 of "y"