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

Determine โซ 9^(log)โ^(๐ฅ) d๐ฅ.

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### Video Transcript

Determine the integral of nine to the power of the logarithm base nine of ๐ฅ with respect to ๐ฅ.

Weโ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 ๐ of ๐ฅ actually means. Itโs defined to be the inverse function of ๐ to the power of ๐ฅ. And if these two functions are inverse of one another, ๐ to the power of the logarithm base ๐ of ๐ฅ will just be equal to ๐ฅ. Because theyโre inverse functions, we just have the identity function. We donโt 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 ๐ฅ is just equal to ๐ฅ. So by using our laws of logarithms, weโve shown the integral given to us in the question is equal to the integral of ๐ฅ with respect to ๐ฅ. And we can evaluate this by using the power rule for integration. We get ๐ฅ squared over two plus ๐ถ.

Therefore, weโve shown the integral of nine to the power of log base nine of ๐ฅ with respect to ๐ฅ is equal to ๐ฅ squared over two plus ๐ถ.