Video Transcript
Determine the integral of nine to the power of the logarithm base nine of ๐ฅ with respect to ๐ฅ.
Weโre 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 ๐ of ๐ฅ actually means. Itโs defined to be the inverse function of ๐ to the power of ๐ฅ. And if these two functions are inverse of one another, ๐ to the power of the logarithm base ๐ of ๐ฅ will just be equal to ๐ฅ. Because theyโre inverse functions, we just have the identity function. We donโt change the value of our input.
So if we set ๐ equal to nine, we see nine to the power of the log base nine of ๐ฅ is just equal to ๐ฅ. So by using our laws of logarithms, weโve shown the integral given to us in the question is equal to the integral of ๐ฅ with respect to ๐ฅ. And we can evaluate this by using the power rule for integration. We get ๐ฅ squared over two plus ๐ถ.
Therefore, weโve shown the integral of nine to the power of log base nine of ๐ฅ with respect to ๐ฅ is equal to ๐ฅ squared over two plus ๐ถ.
