Precalculus Examples

$4^{- \frac{x}{3}} = 0.0052$

Step 1

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

$\ln (4^{- \frac{x}{3}}) = \ln (0.0052)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (4^{- \frac{x}{3}})$ by moving $- \frac{x}{3}$ outside the logarithm.

$- \frac{x}{3} \ln (4) = \ln (0.0052)$

Step 2.2

Combine $\ln (4)$ and $\frac{x}{3}$ .

$- \frac{\ln (4) x}{3} = \ln (0.0052)$

Step 3

Simplify the left side.

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Step 3.1

Reorder factors in $- \frac{\ln (4) x}{3}$ .

$- \frac{x \ln (4)}{3} = \ln (0.0052)$

Step 4

Multiply both sides of the equation by $- \frac{3}{\ln (4)}$ .

$- \frac{3}{\ln (4)} (- \frac{x \ln (4)}{3}) = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $- \frac{3}{\ln (4)} (- \frac{x \ln (4)}{3})$ .

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Step 5.1.1.1

Cancel the common factor of $3$ .

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Step 5.1.1.1.1

Move the leading negative in $- \frac{3}{\ln (4)}$ into the numerator.

$\frac{- 3}{\ln (4)} (- \frac{x \ln (4)}{3}) = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5.1.1.1.2

Move the leading negative in $- \frac{x \ln (4)}{3}$ into the numerator.

$\frac{- 3}{\ln (4)} \cdot \frac{- x \ln (4)}{3} = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{\ln (4)} \cdot \frac{- x \ln (4)}{3} = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{\ln (4)} \cdot \frac{- x \ln (4)}{3} = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{\ln (4)} (- x \ln (4)) = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5.1.1.2

Cancel the common factor of $\ln (4)$ .

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Step 5.1.1.2.1

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

$\frac{- 1}{\ln (4)} (\ln (4) (- x)) = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{\ln (4)} (\ln (4) (- x)) = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5.1.1.2.3

Rewrite the expression.

$- - x = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5.1.1.3

Multiply.

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Step 5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{3}{\ln (4)} \ln (0.0052)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{3}{\ln (4)} \ln (0.0052)$ .

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Step 5.2.1.1

Combine $\ln (0.0052)$ and $\frac{3}{\ln (4)}$ .

$x = - \frac{\ln (0.0052) \cdot 3}{\ln (4)}$

Step 5.2.1.2

Move $3$ to the left of $\ln (0.0052)$ .

$x = - \frac{3 \ln (0.0052)}{\ln (4)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 \ln (0.0052)}{\ln (4)}$

Decimal Form:

$x = 11.38090899 \dots$