Solve for x (5/3)^x=(27/125)

${(\frac{5}{3})}^{x} = (\frac{27}{125})$

Step 1

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

$\ln ({(\frac{5}{3})}^{x}) = \ln (\frac{27}{125})$

Step 2

Expand the left side.

Expand $\ln ({(\frac{5}{3})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{5}{3}) = \ln (\frac{27}{125})$

Rewrite $\ln (\frac{5}{3})$ as $\ln (5) - \ln (3)$ .

$x (\ln (5) - \ln (3)) = \ln (\frac{27}{125})$

Step 3

Simplify the left side.

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

$x \ln (\frac{5}{3}) = \ln (\frac{27}{125})$

Step 4

Divide each term in $x \ln (\frac{5}{3}) = \ln (\frac{27}{125})$ by $\ln (\frac{5}{3})$ and simplify.

Divide each term in $x \ln (\frac{5}{3}) = \ln (\frac{27}{125})$ by $\ln (\frac{5}{3})$ .

$\frac{x \ln (\frac{5}{3})}{\ln (\frac{5}{3})} = \frac{\ln (\frac{27}{125})}{\ln (\frac{5}{3})}$

Simplify the left side.

Cancel the common factor of $\ln (\frac{5}{3})$ .

Cancel the common factor.

$\frac{x \ln (\frac{5}{3})}{\ln (\frac{5}{3})} = \frac{\ln (\frac{27}{125})}{\ln (\frac{5}{3})}$

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{27}{125})}{\ln (\frac{5}{3})}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{27}{125})}{\ln (\frac{5}{3})}$

Decimal Form:

$x = - 3$