What is the relationship between $y = 3^{x}$ $y=3^x$ and $y = {log}_{3} x$ $y=log_3x$ ?

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Answer 1 · 2016-10-22T20:05:43

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

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

Let $f (x) = {log}_{3} (x)$ $f(x) = log_3(x)$ and $f^{- 1} (x) = 3^{x}$ $f^-1(x) = 3^x$

$f (f^{- 1} (x)) = {log}_{3} (3^{x}) = x {log}_{3} (3) = \frac{x log 3}{log 3} = x$ $f(f^-1(x)) = log_3(3^x) = xlog_3(3) =(xlog3)/log3 = x$

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

Hopefully this helps!

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