# What is the relationship between y=3^x and y=log_3x?

Oct 22, 2016

They are inverse functions. We use the following proof to show that particular function is another's inverse.

If $f \left(x\right)$ and ${f}^{-} 1 \left(x\right)$ are inverse functions, then $f \left({f}^{-} 1 \left(x\right)\right) = x$.

Let $f \left(x\right) = {\log}_{3} \left(x\right)$ and ${f}^{-} 1 \left(x\right) = {3}^{x}$

$f \left({f}^{-} 1 \left(x\right)\right) = {\log}_{3} \left({3}^{x}\right) = x {\log}_{3} \left(3\right) = \frac{x \log 3}{\log} 3 = x$

This proves that the relationship between the two functions is that they're inverses.

Hopefully this helps!