Precalculus Examples

$\log_{x} (8) = 0.5$

Step 1

Rewrite $\log_{x} (8) = 0.5$ 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 $b^{y} = x$ .

$x^{0.5} = 8$

Step 2

Solve for $x$ .

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Step 2.1

Convert the decimal exponent to a fractional exponent.

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Step 2.1.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there is $1$ number to the right of the decimal point, place the decimal number over $10^{1}$ $(10)$ . Next, add the whole number to the left of the decimal.

$x^{0 \frac{5}{10}} = 8$

Step 2.1.2

Reduce the fraction.

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Step 2.1.2.1

Convert $0 \frac{5}{10}$ to an improper fraction.

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Step 2.1.2.1.1

A mixed number is an addition of its whole and fractional parts.

$x^{0 + \frac{5}{10}} = 8$

Step 2.1.2.1.2

Add $0$ and $\frac{5}{10}$ .

$x^{\frac{5}{10}} = 8$

Step 2.1.2.2

Cancel the common factor of $5$ and $10$ .

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Step 2.1.2.2.1

Factor $5$ out of $5$ .

$x^{\frac{5 (1)}{10}} = 8$

Step 2.1.2.2.2

Cancel the common factors.

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Step 2.1.2.2.2.1

Factor $5$ out of $10$ .

$x^{\frac{5 \cdot 1}{5 \cdot 2}} = 8$

Step 2.1.2.2.2.2

Cancel the common factor.

$x^{\frac{5 \cdot 1}{5 \cdot 2}} = 8$

Step 2.1.2.2.2.3

Rewrite the expression.

$x^{\frac{1}{2}} = 8$

Step 2.2

Raise each side of the equation to the power of $\frac{1}{0.5}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{\frac{1}{0.5}} = 8^{\frac{1}{0.5}}$

Step 2.3

Simplify the exponent.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify ${(x^{\frac{1}{2}})}^{\frac{1}{0.5}}$ .

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Step 2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{\frac{1}{0.5}}$ .

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Step 2.3.1.1.1.1

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

$x^{\frac{1}{2} \cdot \frac{1}{0.5}} = 8^{\frac{1}{0.5}}$

Step 2.3.1.1.1.2

Cancel the common factor of $0.5$ .

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Step 2.3.1.1.1.2.1

Factor $0.5$ out of $1$ .

$x^{\frac{0.5 (2)}{2} \cdot \frac{1}{0.5}} = 8^{\frac{1}{0.5}}$

Step 2.3.1.1.1.2.2

Cancel the common factor.

$x^{\frac{0.5 \cdot 2}{2} \cdot \frac{1}{0.5}} = 8^{\frac{1}{0.5}}$

Step 2.3.1.1.1.2.3

Rewrite the expression.

$x^{\frac{2}{2}} = 8^{\frac{1}{0.5}}$

Step 2.3.1.1.1.3

Divide $2$ by $2$ .

$x^{1} = 8^{\frac{1}{0.5}}$

Step 2.3.1.1.2

Simplify.

$x = 8^{\frac{1}{0.5}}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $8^{\frac{1}{0.5}}$ .

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Step 2.3.2.1.1

Divide $1$ by $0.5$ .

$x = 8^{2}$

Step 2.3.2.1.2

Raise $8$ to the power of $2$ .

$x = 64$