Solve for x 3^x*9^x=3^15

$3^{x} \cdot 9^{x} = 3^{15}$

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

$\ln (3^{x} \cdot 9^{x}) = \ln (3^{15})$

Expand the left side.

Rewrite $\ln (3^{x} \cdot 9^{x})$ as $\ln (3^{x}) + \ln (9^{x})$ .

$\ln (3^{x}) + \ln (9^{x}) = \ln (3^{15})$

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

$x \ln (3) + \ln (9^{x}) = \ln (3^{15})$

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

$x \ln (3) + x \ln (9) = \ln (3^{15})$

Simplify the right side.

Raise $3$ to the power of $15$ .

$x \ln (3) + x \ln (9) = \ln (14348907)$

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (3) + x \ln (9) - \ln (14348907) = 0$

Add $\ln (14348907)$ to both sides of the equation.

$x \ln (3) + x \ln (9) = \ln (14348907)$

Factor $x$ out of $x \ln (3) + x \ln (9)$ .

Factor $x$ out of $x \ln (3)$ .

$x (\ln (3)) + x \ln (9) = \ln (14348907)$

Factor $x$ out of $x \ln (9)$ .

$x (\ln (3)) + x (\ln (9)) = \ln (14348907)$

Factor $x$ out of $x (\ln (3)) + x (\ln (9))$ .

$x (\ln (3) + \ln (9)) = \ln (14348907)$

Divide each term in $x (\ln (3) + \ln (9)) = \ln (14348907)$ by $\ln (3) + \ln (9)$ and simplify.

Divide each term in $x (\ln (3) + \ln (9)) = \ln (14348907)$ by $\ln (3) + \ln (9)$ .

$\frac{x (\ln (3) + \ln (9))}{\ln (3) + \ln (9)} = \frac{\ln (14348907)}{\ln (3) + \ln (9)}$

Simplify the left side.

Cancel the common factor of $\ln (3) + \ln (9)$ .

Cancel the common factor.

$\frac{x (\ln (3) + \ln (9))}{\ln (3) + \ln (9)} = \frac{\ln (14348907)}{\ln (3) + \ln (9)}$

Divide $x$ by $1$ .

$x = \frac{\ln (14348907)}{\ln (3) + \ln (9)}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (14348907)}{\ln (3) + \ln (9)}$

Decimal Form:

$x = 5$