Solve for x (0.500)^(x/4.50)=0.100

${(0.5)}^{\frac{x}{4.5}} = 0.1$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({0.5}^{\frac{x}{4.5}}) = \ln (0.1)$

Expand the left side.

Expand $\ln ({0.5}^{\frac{x}{4.5}})$ by moving $\frac{x}{4.5}$ outside the logarithm.

$\frac{x}{4.5} \ln (0.5) = \ln (0.1)$

Combine $\frac{x}{4.5}$ and $\ln (0.5)$ .

$\frac{x \ln (0.5)}{4.5} = \ln (0.1)$

Simplify the left side.

Simplify $\frac{x \ln (0.5)}{4.5}$ .

Factor $\ln (0.5)$ out of $x \ln (0.5)$ .

$\frac{\ln (0.5) x}{4.5} = \ln (0.1)$

Factor $4.5$ out of $4.5$ .

$\frac{\ln (0.5) x}{4.5 (1)} = \ln (0.1)$

Separate fractions.

$\frac{\ln (0.5)}{4.5} \cdot \frac{x}{1} = \ln (0.1)$

Replace $e$ with an approximation.

$\frac{\log_{2.71828182} (0.5)}{4.5} \cdot \frac{x}{1} = \ln (0.1)$

Log base $2.71828182$ of $0.5$ is approximately $- 0.69314718$ .

$\frac{- 0.69314718}{4.5} \cdot \frac{x}{1} = \ln (0.1)$

Divide $- 0.69314718$ by $4.5$ .

$- 0.1540327 \frac{x}{1} = \ln (0.1)$

Divide $x$ by $1$ .

$- 0.1540327 x = \ln (0.1)$

Divide each term in $- 0.1540327 x = \ln (0.1)$ by $- 0.1540327$ and simplify.

Divide each term in $- 0.1540327 x = \ln (0.1)$ by $- 0.1540327$ .

$\frac{- 0.1540327 x}{- 0.1540327} = \frac{\ln (0.1)}{- 0.1540327}$

Simplify the left side.

Cancel the common factor of $- 0.1540327$ .

Cancel the common factor.

$\frac{- 0.1540327 x}{- 0.1540327} = \frac{\ln (0.1)}{- 0.1540327}$

Divide $x$ by $1$ .

$x = \frac{\ln (0.1)}{- 0.1540327}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\ln (0.1)}{0.1540327}$

Replace $e$ with an approximation.

$x = - \frac{\log_{2.71828182} (0.1)}{0.1540327}$

Log base $2.71828182$ of $0.1$ is approximately $- 2.30258509$ .

$x = - \frac{- 2.30258509}{0.1540327}$

Divide $- 2.30258509$ by $0.1540327$ .

$x = - - 14.94867642$

Multiply $- 1$ by $- 14.94867642$ .

$x = 14.94867642$