Evaluate 5^x=1/127

$5^{x} = \frac{1}{127}$

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

$\ln (5^{x}) = \ln (\frac{1}{127})$

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

$x \ln (5) = \ln (\frac{1}{127})$

Divide each term in $x \ln (5) = \ln (\frac{1}{127})$ by $\ln (5)$ and simplify.

Divide each term in $x \ln (5) = \ln (\frac{1}{127})$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (\frac{1}{127})}{\ln (5)}$

Simplify the left side.

Cancel the common factor of $\ln (5)$ .

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (\frac{1}{127})}{\ln (5)}$

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{127})}{\ln (5)}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{1}{127})}{\ln (5)}$

Decimal Form:

$x = - 3.00986266 \dots$