Using the properties of logarithms: multiple steps
About this video
Sal simplifies log base 5 of (25^x / y) in several steps. He uses the quotient rule to split it into log_5(25^x) - log_5(y), the power rule to bring the x out front, and then evaluates log_5(25) = 2 to get 2x - log_5(y).
You'll be able to
- use the quotient rule to split the log of a fraction into a difference of logs.
- use the power rule to move an exponent in front of a logarithm.
- evaluate simple logs like log_5(25) and combine the steps to simplify an expression.
Key words
- Quotient rule
- log_x(a/b) = log_x(a) - log_x(b): the log of a quotient is the difference of the logs.
- Power rule
- log_x(a^b) = b·log_x(a): an exponent inside a log can be moved out front as a multiplier.
- Logarithm
- log_b(c) asks 'what power do I raise b to in order to get c?'
Transcript
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 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 one of our logarithm properties-- is logarithm for a given base-- so let's say that the base is x-- of a/b, that is equal to log base x of a minus 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. I'll do 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. 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 be moved out front, which is what we did it right over there.
So this part right over here can be rewritten 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-- and I'll write it in front of the x now-- 2 times x minus log base 5 of y. And we're done.
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