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

Determine ∫ 9^(log)₉^(π‘₯) dπ‘₯.

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Video Transcript

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

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 π‘₯ actually means. It’s defined to be the inverse function of 𝑏 to the power of π‘₯. And if these two functions are inverse of one another, 𝑏 to the power of the logarithm base 𝑏 of π‘₯ will just be equal to π‘₯. 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 π‘₯ is just equal to π‘₯. So by using our laws of logarithms, we’ve 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 𝐢.

Therefore, we’ve 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 𝐢.

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