Question Video: Finding the Integration of an Exponential Function Involving Using the Properties of Exponents and Logarithms Mathematics

Video Transcript

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

We’re 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 𝑏 of 𝑥 actually means. It’s defined to be the inverse function of 𝑏 to the power of 𝑥. And if these two functions are inverse of one another, 𝑏 to the power of the logarithm base 𝑏 of 𝑥 will just be equal to 𝑥. Because they’re inverse functions, we just have the identity function. We don’t change the value of our input.

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

Therefore, we’ve shown the integral of nine to the power of log base nine of 𝑥 with respect to 𝑥 is equal to 𝑥 squared over two plus 𝐶.