### Video Transcript

If π to the power of π¦ is equal to π to the power of π₯ such that π and π are in the real positive numbers and π is not equal to π, find ππ¦ by ππ₯.

In order to find ππ¦ by ππ₯, the first thing weβre going to need to do is to get this equation in terms of π¦. In our equation, π¦ is part of an exponential, as it is in π to the power of π¦. In order to get the π¦ out of the exponential, weβre going to need to perform the inverse operation.

Since we have π to the power of π¦, weβre going to need to take the log of both sides with a base of π. So taking the log to base π of both sides of this equation gives us log to base π of π to the power of π¦ is equal to log to base π of π to the power of π₯.

Our next step will be to use a logarithm rule. This logarithm rule tells us log to base π of π to the power of π is equal to π timesed by the log to base π of π. What this is telling us is that we can take our power and bring it out to the front of the logarithm. And we can do this to π₯ also. This gives us that π¦ timesed by log of base π of π is equal to π₯ timesed by log of base π of π.

Next, weβll be using another logarithm rule. This rule tells us that log to base π of π is equal to one. And so, therefore, log to base π of π is also equal to one. This gives us that π¦ is equal to π₯ multiplied by the log of base π of π. Now we have found π¦ in terms of π₯. And weβre ready to differentiate to find ππ¦ by ππ₯.

We simply differentiate this whole equation with respect to π₯. And this gives us ππ¦ by ππ₯ is equal to log to base π of π. The reason why we get log to base π of π here is because log to base π of π is a constant. And π₯ simply differentiates to one. So we end up with one timesed by the log to base π of π, leaving us with just the log to base π of π. So this is the solution to the question.