Solve for x 5^x=21

$5^{x} = 21$

Step 1

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

$\ln (5^{x}) = \ln (21)$

Step 2

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

$x \ln (5) = \ln (21)$

Step 3

Divide each term in $x \ln (5) = \ln (21)$ by $\ln (5)$ and simplify.

Divide each term in $x \ln (5) = \ln (21)$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (21)}{\ln (5)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (21)}{\ln (5)}$

Divide $x$ by $1$ .

$x = \frac{\ln (21)}{\ln (5)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (21)}{\ln (5)}$

Decimal Form:

$x = 1.89166814 \dots$