# Proof of Log Properties: Rule 2

by Batool Akmal

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DLM Differentiation of Exponential and Logarithmic Functions Calculus Akmal.pdf
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00:01 Let?s move on to our second rule of log.

00:03 So, our second rule of log stated if you have log base a, x over y that is the same as log base a of x minus log base a of y.

00:15 Now, it's very similar to the first rule, we start off doing the same thing makes substitutions for p so this can be p log base a of x and this can be q log base a of y, you can then rewrite this in index form, so let?s just say we're changing this to index form here, so, that becomes a to the p equals to x and that becomes a to the q equals to y.

00:43 So, it's exactly the same step as you did with the first proof, we're now starting our proof with trying to prove this part here, so you can say that x over y, so this is x this is y and x is a to the p, so you can rewrite this is a to the p over a to the q, which if you?re dividing to base numbers you can rewrite this is a to the p minus q.

01:07 So, so far, we can say we have x over y equals to a to the p minus q, you extend this a little bit so you log both sides, so we're going to have log base a of this and log base a of this.

01:22 Looking back at our previous proof, you know that you can bring p minus q down and this entire thing by itself will just equal to one, so, we have log base a, x over y and if you replace your p and q, so, p is log base a of x and q is log base a of y, we can write this as that minus log base a of y, if we just write it out in one line here.

01:48 So, log base a, x over y as the same as log base a of x minus log base a of y.

The lecture Proof of Log Properties: Rule 2 by Batool Akmal is from the course Differentiation of Exponential and Logarithmic Functions.

### Included Quiz Questions

1. log(x/y)
2. x/y
3. log(xy)
4. logᵧ(x) - logₓ(y)
5. log(x) / log(y)
1. 1
2. 0
3. log₁₀(90)
4. -1
5. 9

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