### Video Transcript

Determine the integral of nine to the power of the logarithm base nine of π₯ with respect to π₯.

Weβre asked to evaluate an integral. And we can see that our integrand is in a special form. We need to recall what the logarithm base π of π₯ actually means. Itβs defined to be the inverse function of π to the power of π₯. And if these two functions are inverse of one another, π to the power of the logarithm base π of π₯ will just be equal to π₯. Because theyβre inverse functions, we just have the identity function. We donβt change the value of our input.

So if we set π equal to nine, we see nine to the power of the log base nine of π₯ is just equal to π₯. So by using our laws of logarithms, weβve shown the integral given to us in the question is equal to the integral of π₯ with respect to π₯. And we can evaluate this by using the power rule for integration. We get π₯ squared over two plus πΆ.

Therefore, weβve shown the integral of nine to the power of log base nine of π₯ with respect to π₯ is equal to π₯ squared over two plus πΆ.