Precalculus Examples

$12^{\frac{x}{6}} = 18$

Step 1

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

$\ln (12^{\frac{x}{6}}) = \ln (18)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (12^{\frac{x}{6}})$ by moving $\frac{x}{6}$ outside the logarithm.

$\frac{x}{6} \ln (12) = \ln (18)$

Step 2.2

Combine $\frac{x}{6}$ and $\ln (12)$ .

$\frac{x \ln (12)}{6} = \ln (18)$

Step 3

Multiply both sides of the equation by $\frac{6}{\ln (12)}$ .

$\frac{6}{\ln (12)} \cdot \frac{x \ln (12)}{6} = \frac{6}{\ln (12)} \ln (18)$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{6}{\ln (12)} \cdot \frac{x \ln (12)}{6}$ .

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Step 4.1.1.1

Cancel the common factor of $6$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{6}{\ln (12)} \cdot \frac{x \ln (12)}{6} = \frac{6}{\ln (12)} \ln (18)$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{\ln (12)} (x \ln (12)) = \frac{6}{\ln (12)} \ln (18)$

Step 4.1.1.2

Cancel the common factor of $\ln (12)$ .

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Step 4.1.1.2.1

Factor $\ln (12)$ out of $x \ln (12)$ .

$\frac{1}{\ln (12)} (\ln (12) x) = \frac{6}{\ln (12)} \ln (18)$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{\ln (12)} (\ln (12) x) = \frac{6}{\ln (12)} \ln (18)$

Step 4.1.1.2.3

Rewrite the expression.

$x = \frac{6}{\ln (12)} \ln (18)$

Step 4.2

Simplify the right side.

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Step 4.2.1

Combine $\frac{6}{\ln (12)}$ and $\ln (18)$ .

$x = \frac{6 \ln (18)}{\ln (12)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{6 \ln (18)}{\ln (12)}$

Decimal Form:

$x = 6.97902697 \dots$