Solve for x (1/4)^x=75

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

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

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

Expand the left side.

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

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

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

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

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

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

Subtract $\ln (4)$ from $0$ .

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

Simplify $x (- \ln (4))$ .

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Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Divide each term by $- 1 \ln (4)$ and simplify.

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Divide each term in $- x \ln (4) = \ln (75)$ by $- 1 \ln (4)$ .

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

Simplify $\frac{- x \ln (4)}{- 1 \ln (4)}$ .

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Dividing two negative values results in a positive value.

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

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

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Move the negative in front of the fraction.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 3.11440934 \dots$

Solve for x (1/4)^x=75

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