Pre-Algebra Examples

$- \frac{4}{5^{r}} = - 12$

Step 1

Multiply both sides by $5^{r}$ .

$- \frac{4}{5^{r}} \cdot 5^{r} = - 12 \cdot 5^{r}$

Step 2

Simplify the left side.

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Step 2.1

Cancel the common factor of $5^{r}$ .

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Step 2.1.1

Move the leading negative in $- \frac{4}{5^{r}}$ into the numerator.

$\frac{- 4}{5^{r}} \cdot 5^{r} = - 12 \cdot 5^{r}$

Step 2.1.2

Cancel the common factor.

$\frac{- 4}{5^{r}} \cdot 5^{r} = - 12 \cdot 5^{r}$

Step 2.1.3

Rewrite the expression.

$- 4 = - 12 \cdot 5^{r}$

Step 3

Solve for $r$ .

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Step 3.1

Rewrite the equation as $- 12 \cdot 5^{r} = - 4$ .

$- 12 \cdot 5^{r} = - 4$

Step 3.2

Divide each term in $- 12 \cdot 5^{r} = - 4$ by $- 12$ and simplify.

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Step 3.2.1

Divide each term in $- 12 \cdot 5^{r} = - 4$ by $- 12$ .

$\frac{- 12 \cdot 5^{r}}{- 12} = \frac{- 4}{- 12}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 12$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 12 \cdot 5^{r}}{- 12} = \frac{- 4}{- 12}$

Step 3.2.2.1.2

Divide $5^{r}$ by $1$ .

$5^{r} = \frac{- 4}{- 12}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $- 4$ and $- 12$ .

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Step 3.2.3.1.1

Factor $- 4$ out of $- 4$ .

$5^{r} = \frac{- 4 (1)}{- 12}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $- 4$ out of $- 12$ .

$5^{r} = \frac{- 4 \cdot 1}{- 4 \cdot 3}$

Step 3.2.3.1.2.2

Cancel the common factor.

$5^{r} = \frac{- 4 \cdot 1}{- 4 \cdot 3}$

Step 3.2.3.1.2.3

Rewrite the expression.

$5^{r} = \frac{1}{3}$

Step 3.3

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

$\ln (5^{r}) = \ln (\frac{1}{3})$

Step 3.4

Expand $\ln (5^{r})$ by moving $r$ outside the logarithm.

$r \ln (5) = \ln (\frac{1}{3})$

Step 3.5

Divide each term in $r \ln (5) = \ln (\frac{1}{3})$ by $\ln (5)$ and simplify.

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Step 3.5.1

Divide each term in $r \ln (5) = \ln (\frac{1}{3})$ by $\ln (5)$ .

$\frac{r \ln (5)}{\ln (5)} = \frac{\ln (\frac{1}{3})}{\ln (5)}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $\ln (5)$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{r \ln (5)}{\ln (5)} = \frac{\ln (\frac{1}{3})}{\ln (5)}$

Step 3.5.2.1.2

Divide $r$ by $1$ .

$r = \frac{\ln (\frac{1}{3})}{\ln (5)}$