Precalculus Examples

${(\frac{1}{6})}^{3 x + 2} \cdot 216^{3 x} = \frac{1}{216}$

Step 1

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

$\ln ({(\frac{1}{6})}^{3 x + 2} \cdot 216^{3 x}) = \ln (\frac{1}{216})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln ({(\frac{1}{6})}^{3 x + 2} \cdot 216^{3 x})$ as $\ln ({(\frac{1}{6})}^{3 x + 2}) + \ln (216^{3 x})$ .

$\ln ({(\frac{1}{6})}^{3 x + 2}) + \ln (216^{3 x}) = \ln (\frac{1}{216})$

Step 2.2

Expand $\ln ({(\frac{1}{6})}^{3 x + 2})$ by moving $3 x + 2$ outside the logarithm.

$(3 x + 2) \ln (\frac{1}{6}) + \ln (216^{3 x}) = \ln (\frac{1}{216})$

Step 2.3

Expand $\ln (216^{3 x})$ by moving $3 x$ outside the logarithm.

$(3 x + 2) \ln (\frac{1}{6}) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 2.4

Rewrite $\ln (\frac{1}{6})$ as $\ln (1) - \ln (6)$ .

$(3 x + 2) (\ln (1) - \ln (6)) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 2.5

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

$(3 x + 2) (0 - \ln (6)) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 2.6

Rewrite $\ln (6)$ as $\ln (2 (3))$ .

$(3 x + 2) (0 - \ln (2 (3))) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 2.7

Rewrite $\ln (2 (3))$ as $\ln (2) + \ln (3)$ .

$(3 x + 2) (0 - (\ln (2) + \ln (3))) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 2.8

Apply the distributive property.

$(3 x + 2) (0 - \ln (2) - \ln (3)) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 2.9

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

$(3 x + 2) (- \ln (2) - \ln (3)) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 3

Simplify the left side.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Apply the distributive property.

$3 x (- \ln (2) - \ln (3)) + 2 (- \ln (2) - \ln (3)) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 3.1.1.2

Apply the distributive property.

$3 x (- \ln (2)) + 3 x (- \ln (3)) + 2 (- \ln (2) - \ln (3)) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 3.1.1.3

Apply the distributive property.

$3 x (- \ln (2)) + 3 x (- \ln (3)) + 2 (- \ln (2)) + 2 (- \ln (3)) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Multiply $- 1$ by $3$ .

$- 3 x \ln (2) + 3 x (- \ln (3)) + 2 (- \ln (2)) + 2 (- \ln (3)) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 3.1.2.2

Multiply $- 1$ by $3$ .

$- 3 x \ln (2) - 3 x \ln (3) + 2 (- \ln (2)) + 2 (- \ln (3)) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 3.1.2.3

Multiply $- 1$ by $2$ .

$- 3 x \ln (2) - 3 x \ln (3) - 2 \ln (2) + 2 (- \ln (3)) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 3.1.2.4

Multiply $- 1$ by $2$ .

$- 3 x \ln (2) - 3 x \ln (3) - 2 \ln (2) - 2 \ln (3) + 3 x \ln (216) = \ln (\frac{1}{216})$

Step 4

Simplify the left side.

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Step 4.1

Move $- 2 \ln (3)$ .

$- 3 x \ln (2) - 3 x \ln (3) - 2 \ln (2) + 3 x \ln (216) - 2 \ln (3) = \ln (\frac{1}{216})$

Step 4.2

Move $- 2 \ln (2)$ .

$- 3 x \ln (2) - 3 x \ln (3) + 3 x \ln (216) - 2 \ln (2) - 2 \ln (3) = \ln (\frac{1}{216})$

Step 5

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

$- 3 x \ln (2) - 3 x \ln (3) + 3 x \ln (216) - 2 \ln (2) - 2 \ln (3) - \ln (\frac{1}{216}) = 0$

Step 6

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

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Step 6.1

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

$- 3 x \ln (2) - 3 x \ln (3) + 3 x \ln (216) - 2 \ln (3) - \ln (\frac{1}{216}) = 2 \ln (2)$

Step 6.2

Add $2 \ln (3)$ to both sides of the equation.

$- 3 x \ln (2) - 3 x \ln (3) + 3 x \ln (216) - \ln (\frac{1}{216}) = 2 \ln (2) + 2 \ln (3)$

Step 6.3

Add $\ln (\frac{1}{216})$ to both sides of the equation.

$- 3 x \ln (2) - 3 x \ln (3) + 3 x \ln (216) = 2 \ln (2) + 2 \ln (3) + \ln (\frac{1}{216})$

Step 7

Factor $3 x$ out of $- 3 x \ln (2) - 3 x \ln (3) + 3 x \ln (216)$ .

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Step 7.1

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

$3 x (- 1 \ln (2)) - 3 x \ln (3) + 3 x \ln (216) = 2 \ln (2) + 2 \ln (3) + \ln (\frac{1}{216})$

Step 7.2

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

$3 x (- 1 \ln (2)) + 3 x (- 1 \ln (3)) + 3 x \ln (216) = 2 \ln (2) + 2 \ln (3) + \ln (\frac{1}{216})$

Step 7.3

Factor $3 x$ out of $3 x \ln (216)$ .

$3 x (- 1 \ln (2)) + 3 x (- 1 \ln (3)) + 3 x \ln (216) = 2 \ln (2) + 2 \ln (3) + \ln (\frac{1}{216})$

Step 7.4

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

$3 x (- 1 \ln (2) - 1 \ln (3)) + 3 x \ln (216) = 2 \ln (2) + 2 \ln (3) + \ln (\frac{1}{216})$

Step 7.5

Factor $3 x$ out of $3 x (- 1 \ln (2) - 1 \ln (3)) + 3 x \ln (216)$ .

$3 x (- 1 \ln (2) - 1 \ln (3) + \ln (216)) = 2 \ln (2) + 2 \ln (3) + \ln (\frac{1}{216})$

Step 8

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

$3 x (- \ln (2) - 1 \ln (3) + \ln (216)) = 2 \ln (2) + 2 \ln (3) + \ln (\frac{1}{216})$

Step 9

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

$3 x (- \ln (2) - \ln (3) + \ln (216)) = 2 \ln (2) + 2 \ln (3) + \ln (\frac{1}{216})$

Step 10

Divide each term in $3 x (- \ln (2) - \ln (3) + \ln (216)) = 2 \ln (2) + 2 \ln (3) + \ln (\frac{1}{216})$ by $3 (- \ln (2) - \ln (3) + \ln (216))$ and simplify.

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Step 10.1

Divide each term in $3 x (- \ln (2) - \ln (3) + \ln (216)) = 2 \ln (2) + 2 \ln (3) + \ln (\frac{1}{216})$ by $3 (- \ln (2) - \ln (3) + \ln (216))$ .

$\frac{3 x (- \ln (2) - \ln (3) + \ln (216))}{3 (- \ln (2) - \ln (3) + \ln (216))} = \frac{2 \ln (2)}{3 (- \ln (2) - \ln (3) + \ln (216))} + \frac{2 \ln (3)}{3 (- \ln (2) - \ln (3) + \ln (216))} + \frac{\ln (\frac{1}{216})}{3 (- \ln (2) - \ln (3) + \ln (216))}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $3$ .

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Step 10.2.1.1

Cancel the common factor.

Step 10.2.1.2

Rewrite the expression.

$\frac{x (- \ln (2) - \ln (3) + \ln (216))}{- \ln (2) - \ln (3) + \ln (216)} = \frac{2 \ln (2)}{3 (- \ln (2) - \ln (3) + \ln (216))} + \frac{2 \ln (3)}{3 (- \ln (2) - \ln (3) + \ln (216))} + \frac{\ln (\frac{1}{216})}{3 (- \ln (2) - \ln (3) + \ln (216))}$

Step 10.2.2

Cancel the common factor of $- \ln (2) - \ln (3) + \ln (216)$ .

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Step 10.2.2.1

Cancel the common factor.

Step 10.2.2.2

Divide $x$ by $1$ .

$x = \frac{2 \ln (2)}{3 (- \ln (2) - \ln (3) + \ln (216))} + \frac{2 \ln (3)}{3 (- \ln (2) - \ln (3) + \ln (216))} + \frac{\ln (\frac{1}{216})}{3 (- \ln (2) - \ln (3) + \ln (216))}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 \ln (2)}{3 (- \ln (2) - \ln (3) + \ln (216))} + \frac{2 \ln (3)}{3 (- \ln (2) - \ln (3) + \ln (216))} + \frac{\ln (\frac{1}{216})}{3 (- \ln (2) - \ln (3) + \ln (216))}$

Decimal Form:

$x = - 0.1\overline{6}$