Algebra Examples

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

Step 1

Take the log of both sides of the equation.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Solve the equation for $x$ .

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Simplify the left side.

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Step 6.5.1

Simplify each term.

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Step 6.5.1.1

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

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

Step 6.5.1.2

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

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

Step 6.6

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

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

Step 6.7

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

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Step 6.7.1

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

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

Step 6.7.2

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

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

Step 6.7.3

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

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

Step 6.7.4

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

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

Step 6.7.5

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

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

Step 6.8

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

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Step 6.8.1

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

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

Step 6.8.2

Simplify the left side.

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Step 6.8.2.1

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

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Step 6.8.2.1.1

Cancel the common factor.

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

Step 6.8.2.1.2

Divide $x$ by $1$ .

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

Step 6.8.3

Simplify the right side.

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Step 6.8.3.1

Move the negative in front of the fraction.

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