Finite Math Examples

$- \frac{1}{2} = \log_{x} (9)$

Step 1

Rewrite the equation as $\log_{x} (9) = - \frac{1}{2}$ .

$\log_{x} (9) = - \frac{1}{2}$

Step 2

Rewrite $\log_{x} (9) = - \frac{1}{2}$ 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 $b^{y} = x$ .

$x^{- \frac{1}{2}} = 9$

Step 3

Solve for $x$ .

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Step 3.1

Raise each side of the equation to the power of $- 2$ to eliminate the fractional exponent on the left side.

${(x^{- \frac{1}{2}})}^{- 2} = 9^{- 2}$

Step 3.2

Simplify the exponent.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify ${(x^{- \frac{1}{2}})}^{- 2}$ .

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Step 3.2.1.1.1

Multiply the exponents in ${(x^{- \frac{1}{2}})}^{- 2}$ .

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Step 3.2.1.1.1.1

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

$x^{- \frac{1}{2} \cdot - 2} = 9^{- 2}$

Step 3.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x^{\frac{- 1}{2} \cdot - 2} = 9^{- 2}$

Step 3.2.1.1.1.2.2

Factor $2$ out of $- 2$ .

$x^{\frac{- 1}{2} \cdot (2 (- 1))} = 9^{- 2}$

Step 3.2.1.1.1.2.3

Cancel the common factor.

$x^{\frac{- 1}{2} \cdot (2 \cdot - 1)} = 9^{- 2}$

Step 3.2.1.1.1.2.4

Rewrite the expression.

$x^{- 1 \cdot - 1} = 9^{- 2}$

Step 3.2.1.1.1.3

Multiply $- 1$ by $- 1$ .

$x^{1} = 9^{- 2}$

Step 3.2.1.1.2

Simplify.

$x = 9^{- 2}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $9^{- 2}$ .

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Step 3.2.2.1.1

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

$x = \frac{1}{9^{2}}$

Step 3.2.2.1.2

Raise $9$ to the power of $2$ .

$x = \frac{1}{81}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{81}$

Decimal Form:

$x = 0.01234567 \dots$