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.