Solve for y 9^(2y)=66

$9^{2 y} = 66$

Step 1

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

$\ln (9^{2 y}) = \ln (66)$

Step 2

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

$2 y \ln (9) = \ln (66)$

Step 3

Divide each term in $2 y \ln (9) = \ln (66)$ by $2 \ln (9)$ and simplify.

Divide each term in $2 y \ln (9) = \ln (66)$ by $2 \ln (9)$ .

$\frac{2 y \ln (9)}{2 \ln (9)} = \frac{\ln (66)}{2 \ln (9)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y \ln (9)}{2 \ln (9)} = \frac{\ln (66)}{2 \ln (9)}$

Rewrite the expression.

$\frac{y \ln (9)}{\ln (9)} = \frac{\ln (66)}{2 \ln (9)}$

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

Cancel the common factor.

$\frac{y \ln (9)}{\ln (9)} = \frac{\ln (66)}{2 \ln (9)}$

Divide $y$ by $1$ .

$y = \frac{\ln (66)}{2 \ln (9)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\ln (66)}{2 \ln (9)}$

Decimal Form:

$y = 0.95339702 \dots$