Precalculus Examples

$\log_{x} (256) = - \frac{4}{5}$

Step 1

Rewrite $\log_{x} (256) = - \frac{4}{5}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$x^{- \frac{4}{5}} = 256$

Step 2

Solve for $x$ .

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Step 2.1

Raise each side of the equation to the power of $- \frac{5}{4}$ to eliminate the fractional exponent on the left side.

${(x^{- \frac{4}{5}})}^{- \frac{5}{4}} = \pm 256^{- \frac{5}{4}}$

Step 2.2

Simplify the exponent.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify ${(x^{- \frac{4}{5}})}^{- \frac{5}{4}}$ .

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Step 2.2.1.1.1

Multiply the exponents in ${(x^{- \frac{4}{5}})}^{- \frac{5}{4}}$ .

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Step 2.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{- \frac{4}{5} (- \frac{5}{4})} = \pm 256^{- \frac{5}{4}}$

Step 2.2.1.1.1.2

Cancel the common factor of $4$ .

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Step 2.2.1.1.1.2.1

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

$x^{\frac{- 4}{5} (- \frac{5}{4})} = \pm 256^{- \frac{5}{4}}$

Step 2.2.1.1.1.2.2

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

$x^{\frac{- 4}{5} \cdot \frac{- 5}{4}} = \pm 256^{- \frac{5}{4}}$

Step 2.2.1.1.1.2.3

Factor $4$ out of $- 4$ .

$x^{\frac{4 (- 1)}{5} \cdot \frac{- 5}{4}} = \pm 256^{- \frac{5}{4}}$

Step 2.2.1.1.1.2.4

Cancel the common factor.

$x^{\frac{4 \cdot - 1}{5} \cdot \frac{- 5}{4}} = \pm 256^{- \frac{5}{4}}$

Step 2.2.1.1.1.2.5

Rewrite the expression.

$x^{\frac{- 1}{5} \cdot - 5} = \pm 256^{- \frac{5}{4}}$

Step 2.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 2.2.1.1.1.3.1

Factor $5$ out of $- 5$ .

$x^{\frac{- 1}{5} \cdot (5 (- 1))} = \pm 256^{- \frac{5}{4}}$

Step 2.2.1.1.1.3.2

Cancel the common factor.

$x^{\frac{- 1}{5} \cdot (5 \cdot - 1)} = \pm 256^{- \frac{5}{4}}$

Step 2.2.1.1.1.3.3

Rewrite the expression.

$x^{- - 1} = \pm 256^{- \frac{5}{4}}$

Step 2.2.1.1.1.4

Multiply $- 1$ by $- 1$ .

$x^{1} = \pm 256^{- \frac{5}{4}}$

Step 2.2.1.1.2

Simplify.

$x = \pm 256^{- \frac{5}{4}}$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $\pm 256^{- \frac{5}{4}}$ .

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Step 2.2.2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \pm \frac{1}{256^{\frac{5}{4}}}$

Step 2.2.2.1.2

Simplify the denominator.

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Step 2.2.2.1.2.1

Rewrite $256$ as $4^{4}$ .

$x = \pm \frac{1}{{(4^{4})}^{\frac{5}{4}}}$

Step 2.2.2.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{1}{4^{4 (\frac{5}{4})}}$

Step 2.2.2.1.2.3

Cancel the common factor of $4$ .

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Step 2.2.2.1.2.3.1

Cancel the common factor.

$x = \pm \frac{1}{4^{4 (\frac{5}{4})}}$

Step 2.2.2.1.2.3.2

Rewrite the expression.

$x = \pm \frac{1}{4^{5}}$

Step 2.2.2.1.2.4

Raise $4$ to the power of $5$ .

$x = \pm \frac{1}{1024}$

Step 2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{1}{1024}$

Step 2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{1}{1024}$

Step 2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{1}{1024}, - \frac{1}{1024}$

Step 3

Exclude the solutions that do not make $\log_{x} (256) = - \frac{4}{5}$ true.

$x = \frac{1}{1024}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{1024}$

Decimal Form:

$x = 0.00097656 \dots$