Precalculus Examples

${(\frac{1}{4})}^{x} = 75$

Step 1

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

$\ln ({(\frac{1}{4})}^{x}) = \ln (75)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{1}{4})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{1}{4}) = \ln (75)$

Step 2.2

Rewrite $\ln (\frac{1}{4})$ as $\ln (1) - \ln (4)$ .

$x (\ln (1) - \ln (4)) = \ln (75)$

Step 2.3

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

$x (0 - \ln (4)) = \ln (75)$

Step 2.4

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$x (0 - \ln (2^{2})) = \ln (75)$

Step 2.5

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

$x (0 - (2 \ln (2))) = \ln (75)$

Step 2.6

Multiply $2$ by $- 1$ .

$x (0 - 2 \ln (2)) = \ln (75)$

Step 2.7

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

$x (- 2 \ln (2)) = \ln (75)$

Step 3

Simplify the left side.

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Step 3.1

Reorder factors in $x (- 2 \ln (2))$ .

$- 2 x \ln (2) = \ln (75)$

Step 4

Divide each term in $- 2 x \ln (2) = \ln (75)$ by $- 2 \ln (2)$ and simplify.

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Step 4.1

Divide each term in $- 2 x \ln (2) = \ln (75)$ by $- 2 \ln (2)$ .

$\frac{- 2 x \ln (2)}{- 2 \ln (2)} = \frac{\ln (75)}{- 2 \ln (2)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 2 x \ln (2)}{- 2 \ln (2)} = \frac{\ln (75)}{- 2 \ln (2)}$

Step 4.2.1.2

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (75)}{- 2 \ln (2)}$

Step 4.2.2

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

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Step 4.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (75)}{- 2 \ln (2)}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (75)}{- 2 \ln (2)}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (75)}{2 \ln (2)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (75)}{2 \ln (2)}$

Decimal Form:

$x = - 3.11440934 \dots$