Solve for x 9^(2x)=6

$9^{2 x} = 6$

Step 1

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

$\ln (9^{2 x}) = \ln (6)$

Step 2

Expand $\ln (9^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (9) = \ln (6)$

Step 3

Divide each term in $2 x \ln (9) = \ln (6)$ by $2 \ln (9)$ and simplify.

Divide each term in $2 x \ln (9) = \ln (6)$ by $2 \ln (9)$ .

$\frac{2 x \ln (9)}{2 \ln (9)} = \frac{\ln (6)}{2 \ln (9)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x \ln (9)}{2 \ln (9)} = \frac{\ln (6)}{2 \ln (9)}$

Rewrite the expression.

$\frac{x \ln (9)}{\ln (9)} = \frac{\ln (6)}{2 \ln (9)}$

Cancel the common factor of $\ln (9)$ .

Cancel the common factor.

$\frac{x \ln (9)}{\ln (9)} = \frac{\ln (6)}{2 \ln (9)}$

Divide $x$ by $1$ .

$x = \frac{\ln (6)}{2 \ln (9)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (6)}{2 \ln (9)}$

Decimal Form:

$x = 0.40773243 \dots$