# Is R(x)=4 ln(x) an exponential function?

Jun 26, 2018

$\textcolor{b l u e}{\text{No, it's a logarithmic function}}$

#### Explanation:

$R \left(x\right) = 4 \ln \left(x\right)$ is a logarithmic function.

Logarithmic functions are the inverses of exponential functions.

If we rearrange $y = 4 \ln \left(x\right)$ to find $x$ as a function of $y$

$y = 4 \ln \left(x\right)$

$\frac{y}{4} = \ln \left(x\right)$

${e}^{\frac{y}{4}} = {e}^{\ln \left(x\right)}$

${e}^{\frac{y}{4}} = x$

Substituting $x = y$

$y = {e}^{\frac{x}{4}} \textcolor{w h i t e}{88}$ This is an exponential function.

So:

$y = {e}^{\frac{x}{4}}$ is the inverse of the logarithmic function $y = 4 \ln \left(x\right)$

Conversely: $y = 4 \ln \left(x\right)$ is the inverse of the exponential function $y = {e}^{\frac{x}{4}}$