Algebra Examples

${(\frac{9}{8})}^{3 x + 12} = 1$

Step 1

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

$\ln ({(\frac{9}{8})}^{3 x + 12}) = \ln (1)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{9}{8})}^{3 x + 12})$ by moving $3 x + 12$ outside the logarithm.

$(3 x + 12) \ln (\frac{9}{8}) = \ln (1)$

Step 2.2

Rewrite $\ln (\frac{9}{8})$ as $\ln (9) - \ln (8)$ .

$(3 x + 12) (\ln (9) - \ln (8)) = \ln (1)$

Step 2.3

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$(3 x + 12) (\ln (9) - \ln (2^{3})) = \ln (1)$

Step 2.4

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

$(3 x + 12) (\ln (9) - (3 \ln (2))) = \ln (1)$

Step 2.5

Multiply $3$ by $- 1$ .

$(3 x + 12) (\ln (9) - 3 \ln (2)) = \ln (1)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(3 x + 12) (\ln (9) - 3 \ln (2))$ .

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Step 3.1.1

Expand $(3 x + 12) (\ln (9) - 3 \ln (2))$ using the FOIL Method.

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Step 3.1.1.1

Apply the distributive property.

$3 x (\ln (9) - 3 \ln (2)) + 12 (\ln (9) - 3 \ln (2)) = \ln (1)$

Step 3.1.1.2

Apply the distributive property.

$3 x \ln (9) + 3 x (- 3 \ln (2)) + 12 (\ln (9) - 3 \ln (2)) = \ln (1)$

Step 3.1.1.3

Apply the distributive property.

$3 x \ln (9) + 3 x (- 3 \ln (2)) + 12 \ln (9) + 12 (- 3 \ln (2)) = \ln (1)$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Multiply $- 3$ by $3$ .

$3 x \ln (9) - 9 x \ln (2) + 12 \ln (9) + 12 (- 3 \ln (2)) = \ln (1)$

Step 3.1.2.2

Multiply $- 3$ by $12$ .

$3 x \ln (9) - 9 x \ln (2) + 12 \ln (9) - 36 \ln (2) = \ln (1)$

Step 4

Simplify the right side.

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Step 4.1

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

$3 x \ln (9) - 9 x \ln (2) + 12 \ln (9) - 36 \ln (2) = 0$

Step 5

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1

Subtract $12 \ln (9)$ from both sides of the equation.

$3 x \ln (9) - 9 x \ln (2) - 36 \ln (2) = - 12 \ln (9)$

Step 5.2

Add $36 \ln (2)$ to both sides of the equation.

$3 x \ln (9) - 9 x \ln (2) = - 12 \ln (9) + 36 \ln (2)$

Step 6

Factor $3 x$ out of $3 x \ln (9) - 9 x \ln (2)$ .

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Step 6.1

Factor $3 x$ out of $- 9 x \ln (2)$ .

$3 x \ln (9) + 3 x (- 3 \ln (2)) = - 12 \ln (9) + 36 \ln (2)$

Step 6.2

Factor $3 x$ out of $3 x \ln (9) + 3 x (- 3 \ln (2))$ .

$3 x (\ln (9) - 3 \ln (2)) = - 12 \ln (9) + 36 \ln (2)$

Step 7

Divide each term in $3 x (\ln (9) - 3 \ln (2)) = - 12 \ln (9) + 36 \ln (2)$ by $3 (\ln (9) - 3 \ln (2))$ and simplify.

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Step 7.1

Divide each term in $3 x (\ln (9) - 3 \ln (2)) = - 12 \ln (9) + 36 \ln (2)$ by $3 (\ln (9) - 3 \ln (2))$ .

$\frac{3 x (\ln (9) - 3 \ln (2))}{3 (\ln (9) - 3 \ln (2))} = \frac{- 12 \ln (9)}{3 (\ln (9) - 3 \ln (2))} + \frac{36 \ln (2)}{3 (\ln (9) - 3 \ln (2))}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{3 x (\ln (9) - 3 \ln (2))}{3 (\ln (9) - 3 \ln (2))} = \frac{- 12 \ln (9)}{3 (\ln (9) - 3 \ln (2))} + \frac{36 \ln (2)}{3 (\ln (9) - 3 \ln (2))}$

Step 7.2.1.2

Rewrite the expression.

$\frac{x (\ln (9) - 3 \ln (2))}{\ln (9) - 3 \ln (2)} = \frac{- 12 \ln (9)}{3 (\ln (9) - 3 \ln (2))} + \frac{36 \ln (2)}{3 (\ln (9) - 3 \ln (2))}$

Step 7.2.2

Cancel the common factor of $\ln (9) - 3 \ln (2)$ .

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Step 7.2.2.1

Cancel the common factor.

$\frac{x (\ln (9) - 3 \ln (2))}{\ln (9) - 3 \ln (2)} = \frac{- 12 \ln (9)}{3 (\ln (9) - 3 \ln (2))} + \frac{36 \ln (2)}{3 (\ln (9) - 3 \ln (2))}$

Step 7.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 12 \ln (9)}{3 (\ln (9) - 3 \ln (2))} + \frac{36 \ln (2)}{3 (\ln (9) - 3 \ln (2))}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $- 12$ and $3$ .

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Step 7.3.1.1.1

Factor $3$ out of $- 12 \ln (9)$ .

$x = \frac{3 (- 4 \ln (9))}{3 (\ln (9) - 3 \ln (2))} + \frac{36 \ln (2)}{3 (\ln (9) - 3 \ln (2))}$

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

Cancel the common factor.

$x = \frac{3 (- 4 \ln (9))}{3 (\ln (9) - 3 \ln (2))} + \frac{36 \ln (2)}{3 (\ln (9) - 3 \ln (2))}$

Step 7.3.1.1.2.2

Rewrite the expression.

$x = \frac{- 4 \ln (9)}{\ln (9) - 3 \ln (2)} + \frac{36 \ln (2)}{3 (\ln (9) - 3 \ln (2))}$

Step 7.3.1.2

Move the negative in front of the fraction.

$x = - \frac{4 \ln (9)}{\ln (9) - 3 \ln (2)} + \frac{36 \ln (2)}{3 (\ln (9) - 3 \ln (2))}$

Step 7.3.1.3

Cancel the common factor of $36$ and $3$ .

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Step 7.3.1.3.1

Factor $3$ out of $36 \ln (2)$ .

$x = - \frac{4 \ln (9)}{\ln (9) - 3 \ln (2)} + \frac{3 (12 \ln (2))}{3 (\ln (9) - 3 \ln (2))}$

Step 7.3.1.3.2

Cancel the common factors.

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Step 7.3.1.3.2.1

Cancel the common factor.

$x = - \frac{4 \ln (9)}{\ln (9) - 3 \ln (2)} + \frac{3 (12 \ln (2))}{3 (\ln (9) - 3 \ln (2))}$

Step 7.3.1.3.2.2

Rewrite the expression.

$x = - \frac{4 \ln (9)}{\ln (9) - 3 \ln (2)} + \frac{12 \ln (2)}{\ln (9) - 3 \ln (2)}$

Step 7.3.2

Simplify terms.

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Step 7.3.2.1

Combine the numerators over the common denominator.

$x = \frac{- 4 \ln (9) + 12 \ln (2)}{\ln (9) - 3 \ln (2)}$

Step 7.3.2.2

Factor $4$ out of $- 4 \ln (9) + 12 \ln (2)$ .

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Step 7.3.2.2.1

Factor $4$ out of $- 4 \ln (9)$ .

$x = \frac{4 (- \ln (9)) + 12 \ln (2)}{\ln (9) - 3 \ln (2)}$

Step 7.3.2.2.2

Factor $4$ out of $12 \ln (2)$ .

$x = \frac{4 (- \ln (9)) + 4 (3 \ln (2))}{\ln (9) - 3 \ln (2)}$

Step 7.3.2.2.3

Factor $4$ out of $4 (- \ln (9)) + 4 (3 \ln (2))$ .

$x = \frac{4 (- \ln (9) + 3 \ln (2))}{\ln (9) - 3 \ln (2)}$

Step 7.3.2.3

Cancel the common factor of $- \ln (9) + 3 \ln (2)$ and $\ln (9) - 3 \ln (2)$ .

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Step 7.3.2.3.1

Factor $- 1$ out of $- \ln (9)$ .

$x = \frac{4 (- \ln (9) + 3 \ln (2))}{\ln (9) - 3 \ln (2)}$

Step 7.3.2.3.2

Factor $- 1$ out of $3 \ln (2)$ .

$x = \frac{4 (- \ln (9) - (- 3 \ln (2)))}{\ln (9) - 3 \ln (2)}$

Step 7.3.2.3.3

Factor $- 1$ out of $- \ln (9) - (- 3 \ln (2))$ .

$x = \frac{4 (- (\ln (9) - 3 \ln (2)))}{\ln (9) - 3 \ln (2)}$

Step 7.3.2.3.4

Rewrite $- (\ln (9) - 3 \ln (2))$ as $- 1 (\ln (9) - 3 \ln (2))$ .

$x = \frac{4 (- 1 (\ln (9) - 3 \ln (2)))}{\ln (9) - 3 \ln (2)}$

Step 7.3.2.3.5

Cancel the common factor.

$x = \frac{4 (- 1 (\ln (9) - 3 \ln (2)))}{\ln (9) - 3 \ln (2)}$

Step 7.3.2.3.6

Divide $4 \cdot (- 1)$ by $1$ .

$x = 4 \cdot (- 1)$

Step 7.3.2.4

Multiply $4$ by $- 1$ .

$x = - 4$