Finite Math Examples

$\log_{x} (32) = - \frac{5}{3}$

Step 1

Rewrite $\log_{x} (32) = - \frac{5}{3}$ 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 $b^{y} = x$ .

$x^{- \frac{5}{3}} = 32$

Step 2

Solve for $x$ .

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Step 2.1

Raise each side of the equation to the power of $- \frac{3}{5}$ to eliminate the fractional exponent on the left side.

${(x^{- \frac{5}{3}})}^{- \frac{3}{5}} = 32^{- \frac{3}{5}}$

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 ${(x^{- \frac{5}{3}})}^{- \frac{3}{5}}$ .

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Step 2.2.1.1.1

Multiply the exponents in ${(x^{- \frac{5}{3}})}^{- \frac{3}{5}}$ .

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Step 2.2.1.1.1.1

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

$x^{- \frac{5}{3} (- \frac{3}{5})} = 32^{- \frac{3}{5}}$

Step 2.2.1.1.1.2

Cancel the common factor of $5$ .

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Step 2.2.1.1.1.2.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$x^{\frac{- 5}{3} (- \frac{3}{5})} = 32^{- \frac{3}{5}}$

Step 2.2.1.1.1.2.2

Move the leading negative in $- \frac{3}{5}$ into the numerator.

$x^{\frac{- 5}{3} \cdot \frac{- 3}{5}} = 32^{- \frac{3}{5}}$

Step 2.2.1.1.1.2.3

Factor $5$ out of $- 5$ .

$x^{\frac{5 (- 1)}{3} \cdot \frac{- 3}{5}} = 32^{- \frac{3}{5}}$

Step 2.2.1.1.1.2.4

Cancel the common factor.

$x^{\frac{5 \cdot - 1}{3} \cdot \frac{- 3}{5}} = 32^{- \frac{3}{5}}$

Step 2.2.1.1.1.2.5

Rewrite the expression.

$x^{\frac{- 1}{3} \cdot - 3} = 32^{- \frac{3}{5}}$

Step 2.2.1.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.1.3.1

Factor $3$ out of $- 3$ .

$x^{\frac{- 1}{3} \cdot (3 (- 1))} = 32^{- \frac{3}{5}}$

Step 2.2.1.1.1.3.2

Cancel the common factor.

$x^{\frac{- 1}{3} \cdot (3 \cdot - 1)} = 32^{- \frac{3}{5}}$

Step 2.2.1.1.1.3.3

Rewrite the expression.

$x^{- - 1} = 32^{- \frac{3}{5}}$

Step 2.2.1.1.1.4

Multiply $- 1$ by $- 1$ .

$x^{1} = 32^{- \frac{3}{5}}$

Step 2.2.1.1.2

Simplify.

$x = 32^{- \frac{3}{5}}$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $32^{- \frac{3}{5}}$ .

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Step 2.2.2.1.1

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

$x = \frac{1}{32^{\frac{3}{5}}}$

Step 2.2.2.1.2

Simplify the denominator.

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Step 2.2.2.1.2.1

Rewrite $32$ as $2^{5}$ .

$x = \frac{1}{{(2^{5})}^{\frac{3}{5}}}$

Step 2.2.2.1.2.2

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

$x = \frac{1}{2^{5 (\frac{3}{5})}}$

Step 2.2.2.1.2.3

Cancel the common factor of $5$ .

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Step 2.2.2.1.2.3.1

Cancel the common factor.

$x = \frac{1}{2^{5 (\frac{3}{5})}}$

Step 2.2.2.1.2.3.2

Rewrite the expression.

$x = \frac{1}{2^{3}}$

Step 2.2.2.1.2.4

Raise $2$ to the power of $3$ .

$x = \frac{1}{8}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{8}$

Decimal Form:

$x = 0.125$