How do you find the inverse of f(x)= -log_5 (x-3)?

1 Answer
Jan 2, 2016

Remember Inverse of $\left(x , y\right)$ is given by $\left(y , x\right)$ we are going to use the same to find the inverse. Step by step procedure is given below.

Explanation:

Inverse of $\left(x , y\right)$ is $\left(y , x\right)$

Our process starts by swapping $x$ and $y$

$f \left(x\right) = - {\log}_{5} \left(x - 3\right)$

Remember $y$ and $f \left(x\right)$ are the same.

$y = - {\log}_{5} \left(x - 3\right)$

Step 1: Swap $x$ and $y$

$x = - {\log}_{5} \left(y - 3\right)$

Multiply both sides by $- 1$

$- x = {\log}_{5} \left(y - 3\right)$

Step 2: Solve for $y$

This requires you to have some knowledge on converting log to exponent form.

${\log}_{b} \left(a\right) = k \implies a = {b}^{k}$

${\log}_{5} \left(y - 3\right) = - x$

$y - 3 = {5}^{-} x$

Solving for $y$.

Adding $3$ on both sides would do the trick here.

$y = {5}^{-} x + 3$

This $y$ is the inverse function and to be represented as ${f}^{-} 1 \left(x\right)$

Our answer ${f}^{-} 1 \left(x\right) = {5}^{-} x + 3$