Algebra Examples

$4 \log_{x} (\sqrt{2}) = \frac{1}{2}$

Step 1

Divide each term in $4 \log_{x} (\sqrt{2}) = \frac{1}{2}$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 \log_{x} (\sqrt{2}) = \frac{1}{2}$ by $4$ .

$\frac{4 \log_{x} (\sqrt{2})}{4} = \frac{\frac{1}{2}}{4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{4 \log_{x} (\sqrt{2})}{4} = \frac{\frac{1}{2}}{4}$

Step 1.2.1.2

Divide $\log_{x} (\sqrt{2})$ by $1$ .

$\log_{x} (\sqrt{2}) = \frac{\frac{1}{2}}{4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Multiply the numerator by the reciprocal of the denominator.

$\log_{x} (\sqrt{2}) = \frac{1}{2} \cdot \frac{1}{4}$

Step 1.3.2

Multiply $\frac{1}{2} \cdot \frac{1}{4}$ .

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Step 1.3.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$\log_{x} (\sqrt{2}) = \frac{1}{2 \cdot 4}$

Step 1.3.2.2

Multiply $2$ by $4$ .

$\log_{x} (\sqrt{2}) = \frac{1}{8}$

Step 2

Rewrite $\log_{x} (\sqrt{2}) = \frac{1}{8}$ 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 $b^{y} = x$ .

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

Step 3

Solve for $x$ .

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Step 3.1

Raise each side of the equation to the power of $8$ to eliminate the fractional exponent on the left side.

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

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 ${(x^{\frac{1}{8}})}^{8}$ .

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Step 3.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{8}})}^{8}$ .

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Step 3.2.1.1.1.1

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

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

Step 3.2.1.1.1.2

Cancel the common factor of $8$ .

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Step 3.2.1.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.1.2

Simplify.

$x = {\sqrt{2}}^{8}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify ${\sqrt{2}}^{8}$ .

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Step 3.2.2.1.1

Rewrite ${\sqrt{2}}^{8}$ as $2^{4}$ .

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Step 3.2.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.2.2.1.1.2

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

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

Step 3.2.2.1.1.3

Combine $\frac{1}{2}$ and $8$ .

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

Step 3.2.2.1.1.4

Cancel the common factor of $8$ and $2$ .

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Step 3.2.2.1.1.4.1

Factor $2$ out of $8$ .

$x = 2^{\frac{2 \cdot 4}{2}}$

Step 3.2.2.1.1.4.2

Cancel the common factors.

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Step 3.2.2.1.1.4.2.1

Factor $2$ out of $2$ .

$x = 2^{\frac{2 \cdot 4}{2 (1)}}$

Step 3.2.2.1.1.4.2.2

Cancel the common factor.

$x = 2^{\frac{2 \cdot 4}{2 \cdot 1}}$

Step 3.2.2.1.1.4.2.3

Rewrite the expression.

$x = 2^{\frac{4}{1}}$

Step 3.2.2.1.1.4.2.4

Divide $4$ by $1$ .

$x = 2^{4}$

Step 3.2.2.1.2

Raise $2$ to the power of $4$ .

$x = 16$