Algebra Examples

${(\frac{1}{9})}^{x} \cdot {(\frac{1}{3})}^{x} = 3^{16 - x}$

Step 1

Take the log of both sides of the equation.

$\ln ({(\frac{1}{9})}^{x} \cdot {(\frac{1}{3})}^{x}) = \ln (3^{16 - x})$

Step 2

Rewrite $\ln ({(\frac{1}{9})}^{x} \cdot {(\frac{1}{3})}^{x})$ as $\ln ({(\frac{1}{9})}^{x}) + \ln ({(\frac{1}{3})}^{x})$ .

$\ln ({(\frac{1}{9})}^{x}) + \ln ({(\frac{1}{3})}^{x}) = \ln (3^{16 - x})$

Step 3

Expand $\ln ({(\frac{1}{9})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{1}{9}) + \ln ({(\frac{1}{3})}^{x}) = \ln (3^{16 - x})$

Step 4

Expand $\ln ({(\frac{1}{3})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{1}{9}) + x \ln (\frac{1}{3}) = \ln (3^{16 - x})$

Step 5

Rewrite $\ln (\frac{1}{9})$ as $\ln (1) - \ln (9)$ .

$x (\ln (1) - \ln (9)) + x \ln (\frac{1}{3}) = \ln (3^{16 - x})$

Step 6

The natural logarithm of $1$ is $0$ .

$x (0 - \ln (9)) + x \ln (\frac{1}{3}) = \ln (3^{16 - x})$

Step 7

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

$x (0 - \ln (3^{2})) + x \ln (\frac{1}{3}) = \ln (3^{16 - x})$

Step 8

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

$x (0 - (2 \ln (3))) + x \ln (\frac{1}{3}) = \ln (3^{16 - x})$

Step 9

Multiply $2$ by $- 1$ .

$x (0 - 2 \ln (3)) + x \ln (\frac{1}{3}) = \ln (3^{16 - x})$

Step 10

Subtract $2 \ln (3)$ from $0$ .

$x (- 2 \ln (3)) + x \ln (\frac{1}{3}) = \ln (3^{16 - x})$

Step 11

Rewrite $\ln (\frac{1}{3})$ as $\ln (1) - \ln (3)$ .

$x (- 2 \ln (3)) + x (\ln (1) - \ln (3)) = \ln (3^{16 - x})$

Step 12

The natural logarithm of $1$ is $0$ .

$x (- 2 \ln (3)) + x (0 - \ln (3)) = \ln (3^{16 - x})$

Step 13

Subtract $\ln (3)$ from $0$ .

$x (- 2 \ln (3)) + x (- \ln (3)) = \ln (3^{16 - x})$

Step 14

Expand $\ln (3^{16 - x})$ by moving $16 - x$ outside the logarithm.

$x (- 2 \ln (3)) + x (- \ln (3)) = (16 - x) \ln (3)$

Step 15

Solve the equation for $x$ .

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Step 15.1

Simplify the left side.

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Step 15.1.1

Reorder $x$ and $- 1$ .

$- 2 \cdot (x \ln (3)) - 1 \cdot (x \ln (3)) = (16 - x) \ln (3)$

Step 15.1.2

Rewrite $- 1 x$ as $- x$ .

$- 2 x \ln (3) - x \ln (3) = (16 - x) \ln (3)$

Step 15.1.3

Subtract $x \ln (3)$ from $- 2 x \ln (3)$ .

$- 3 x \ln (3) = (16 - x) \ln (3)$

Step 15.2

Simplify the right side.

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Step 15.2.1

Apply the distributive property.

$- 3 x \ln (3) = 16 \ln (3) - x \ln (3)$

Step 15.2.2

Reorder $16 \ln (3)$ and $- x \ln (3)$ .

$- 3 x \ln (3) = - x \ln (3) + 16 \ln (3)$

Step 15.3

Simplify the left side.

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Step 15.3.1

Simplify $- 3 x \ln (3)$ .

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Step 15.3.1.1

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

$x (- \ln (3^{3})) = - x \ln (3) + 16 \ln (3)$

Step 15.3.1.2

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

$x (- \ln (27)) = - x \ln (3) + 16 \ln (3)$

Step 15.3.1.3

Reorder factors in $x (- \ln (27))$ .

$- x \ln (27) = - x \ln (3) + 16 \ln (3)$

Step 15.4

Simplify the right side.

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Step 15.4.1

Simplify each term.

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Step 15.4.1.1

Simplify $16 \ln (3)$ by moving $16$ inside the logarithm.

$- x \ln (27) = - x \ln (3) + \ln (3^{16})$

Step 15.4.1.2

Raise $3$ to the power of $16$ .

$- x \ln (27) = - x \ln (3) + \ln (43046721)$

Step 15.5

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

$- x \ln (27) + x \ln (3) - \ln (43046721) = 0$

Step 15.6

Add $\ln (43046721)$ to both sides of the equation.

$- x \ln (27) + x \ln (3) = \ln (43046721)$

Step 15.7

Factor $x$ out of $- x \ln (27) + x \ln (3)$ .

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Step 15.7.1

Factor $x$ out of $- x \ln (27)$ .

$x (- 1 \ln (27)) + x \ln (3) = \ln (43046721)$

Step 15.7.2

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

$x (- 1 \ln (27)) + x (\ln (3)) = \ln (43046721)$

Step 15.7.3

Factor $x$ out of $x (- 1 \ln (27)) + x (\ln (3))$ .

$x (- 1 \ln (27) + \ln (3)) = \ln (43046721)$

Step 15.8

Rewrite $- 1 \ln (27)$ as $- \ln (27)$ .

$x (- \ln (27) + \ln (3)) = \ln (43046721)$

Step 15.9

Divide each term in $x (- \ln (27) + \ln (3)) = \ln (43046721)$ by $- \ln (27) + \ln (3)$ and simplify.

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Step 15.9.1

Divide each term in $x (- \ln (27) + \ln (3)) = \ln (43046721)$ by $- \ln (27) + \ln (3)$ .

$\frac{x (- \ln (27) + \ln (3))}{- \ln (27) + \ln (3)} = \frac{\ln (43046721)}{- \ln (27) + \ln (3)}$

Step 15.9.2

Simplify the left side.

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Step 15.9.2.1

Cancel the common factor of $- \ln (27) + \ln (3)$ .

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Step 15.9.2.1.1

Cancel the common factor.

$\frac{x (- \ln (27) + \ln (3))}{- \ln (27) + \ln (3)} = \frac{\ln (43046721)}{- \ln (27) + \ln (3)}$

Step 15.9.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (43046721)}{- \ln (27) + \ln (3)}$

Step 15.9.3

Simplify the right side.

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Step 15.9.3.1

Factor $- 1$ out of $- \ln (27)$ .

$x = \frac{\ln (43046721)}{- \ln (27) + \ln (3)}$

Step 15.9.3.2

Factor $- 1$ out of $\ln (3)$ .

$x = \frac{\ln (43046721)}{- \ln (27) - 1 (- \ln (3))}$

Step 15.9.3.3

Factor $- 1$ out of $- \ln (27) - 1 (- \ln (3))$ .

$x = \frac{\ln (43046721)}{- (\ln (27) - \ln (3))}$

Step 15.9.3.4

Rewrite negatives.

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Step 15.9.3.4.1

Rewrite $- (\ln (27) - \ln (3))$ as $- 1 (\ln (27) - \ln (3))$ .

$x = \frac{\ln (43046721)}{- 1 (\ln (27) - \ln (3))}$

Step 15.9.3.4.2

Move the negative in front of the fraction.

$x = - \frac{\ln (43046721)}{\ln (27) - \ln (3)}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (43046721)}{\ln (27) - \ln (3)}$

Decimal Form:

$x = - 8$