Algebra Examples

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

Step 1

Rewrite the equation as $x^{a} = \frac{x^{\frac{1}{2}}}{\sqrt{x}}$ .

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

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

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

Step 3

Expand $\ln (x^{a})$ by moving $a$ outside the logarithm.

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

Step 4

Simplify the right side.

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Step 4.1

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

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Step 4.1.1

Multiply $\frac{x^{\frac{1}{2}}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$a \ln (x) = \ln (\frac{x^{\frac{1}{2}}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}})$

Step 4.1.2

Combine and simplify the denominator.

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Step 4.1.2.1

Multiply $\frac{x^{\frac{1}{2}}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 4.1.2.2

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 4.1.2.3

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 4.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a \ln (x) = \ln (\frac{x^{\frac{1}{2}} \sqrt{x}}{{\sqrt{x}}^{1 + 1}})$

Step 4.1.2.5

Add $1$ and $1$ .

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

Step 4.1.2.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 4.1.2.6.1

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

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

Step 4.1.2.6.2

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

$a \ln (x) = \ln (\frac{x^{\frac{1}{2}} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}})$

Step 4.1.2.6.3

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

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

Step 4.1.2.6.4

Cancel the common factor of $2$ .

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Step 4.1.2.6.4.1

Cancel the common factor.

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

Step 4.1.2.6.4.2

Rewrite the expression.

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

Step 4.1.2.6.5

Simplify.

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

Step 4.1.3

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$a \ln (x) = \ln (\frac{\sqrt{x}}{x \cdot x^{- \frac{1}{2}}})$

Step 4.1.4

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.1.4.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 4.1.4.1.1

Raise $x$ to the power of $1$ .

$a \ln (x) = \ln (\frac{\sqrt{x}}{x^{1} x^{- \frac{1}{2}}})$

Step 4.1.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a \ln (x) = \ln (\frac{\sqrt{x}}{x^{1 - \frac{1}{2}}})$

Step 4.1.4.2

Write $1$ as a fraction with a common denominator.

$a \ln (x) = \ln (\frac{\sqrt{x}}{x^{\frac{2}{2} - \frac{1}{2}}})$

Step 4.1.4.3

Combine the numerators over the common denominator.

$a \ln (x) = \ln (\frac{\sqrt{x}}{x^{\frac{2 - 1}{2}}})$

Step 4.1.4.4

Subtract $1$ from $2$ .

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

Step 5

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

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Step 5.1

Divide each term in $a \ln (x) = \ln (\frac{\sqrt{x}}{x^{\frac{1}{2}}})$ by $\ln (x)$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $\ln (x)$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $a$ by $1$ .

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