Solve for x (2/5)^x=(125/8)

${(\frac{2}{5})}^{x} = (\frac{125}{8})$

Step 1

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

$\ln ({(\frac{2}{5})}^{x}) = \ln (\frac{125}{8})$

Step 2

Expand the left side.

Expand $\ln ({(\frac{2}{5})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{2}{5}) = \ln (\frac{125}{8})$

Rewrite $\ln (\frac{2}{5})$ as $\ln (2) - \ln (5)$ .

$x (\ln (2) - \ln (5)) = \ln (\frac{125}{8})$

Step 3

Simplify the left side.

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (\frac{2}{5}) = \ln (\frac{125}{8})$

Step 4

Divide each term in $x \ln (\frac{2}{5}) = \ln (\frac{125}{8})$ by $\ln (\frac{2}{5})$ and simplify.

Divide each term in $x \ln (\frac{2}{5}) = \ln (\frac{125}{8})$ by $\ln (\frac{2}{5})$ .

$\frac{x \ln (\frac{2}{5})}{\ln (\frac{2}{5})} = \frac{\ln (\frac{125}{8})}{\ln (\frac{2}{5})}$

Simplify the left side.

Cancel the common factor of $\ln (\frac{2}{5})$ .

Cancel the common factor.

$\frac{x \ln (\frac{2}{5})}{\ln (\frac{2}{5})} = \frac{\ln (\frac{125}{8})}{\ln (\frac{2}{5})}$

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{125}{8})}{\ln (\frac{2}{5})}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{125}{8})}{\ln (\frac{2}{5})}$

Decimal Form:

$x = - 3$