Solve the Rational Equation for x 9^x*3^(x+2)=3^(f(x))

$9^{x} \cdot 3^{x + 2} = 3^{f (x)}$

Take the log of both sides of the equation.

$\ln (9^{x} \cdot 3^{x + 2}) = \ln (3^{f (x)})$

Rewrite $\ln (9^{x} \cdot 3^{x + 2})$ as $\ln (9^{x}) + \ln (3^{x + 2})$ .

$\ln (9^{x}) + \ln (3^{x + 2}) = \ln (3^{f (x)})$

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

$x \ln (9) + \ln (3^{x + 2}) = \ln (3^{f (x)})$

Expand $\ln (3^{x + 2})$ by moving $x + 2$ outside the logarithm.

$x \ln (9) + (x + 2) \ln (3) = \ln (3^{f (x)})$

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

$x \ln (9) + (x + 2) \ln (3) = f (x) \ln (3)$

Solve the equation for $x$ .

Apply the distributive property.

$x \ln (9) + x \ln (3) + 2 \ln (3) = f (x) \ln (3)$

Reorder factors in $f (x) \ln (3)$ .

$x \ln (9) + x \ln (3) + 2 \ln (3) = \ln (3) f (x)$

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

$x \ln (9) + x \ln (3) + 2 \ln (3) - \ln (3) f x = 0$

Reorder factors in $x \ln (9) + x \ln (3) + 2 \ln (3) - \ln (3) f x$ .

$x \ln (9) + x \ln (3) + 2 \ln (3) - f x \ln (3) = 0$

Simplify the left side.

Simplify each term.

Simplify $2 \ln (3)$ by moving $2$ inside the logarithm.

$x \ln (9) + x \ln (3) + \ln (3^{2}) - f x \ln (3) = 0$

Raise $3$ to the power of $2$ .

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

Subtract $\ln (9)$ from both sides of the equation.

$x \ln (9) + x \ln (3) - f x \ln (3) = - \ln (9)$

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

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

$x (\ln (9)) + x \ln (3) - f x \ln (3) = - \ln (9)$

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

$x (\ln (9)) + x (\ln (3)) - f x \ln (3) = - \ln (9)$

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

$x (\ln (9)) + x (\ln (3)) + x (- f \ln (3)) = - \ln (9)$

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

$x (\ln (9) + \ln (3)) + x (- f \ln (3)) = - \ln (9)$

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

$x (\ln (9) + \ln (3) - f \ln (3)) = - \ln (9)$

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

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

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

Simplify the left side.

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

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Solve the Rational Equation for x 9^x*3^(x+2)=3^(f(x))

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