Precalculus Examples

$\ln (\sqrt{x}) - \frac{1}{5} \cdot \ln (x) + \ln (\sqrt[4]{x})$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$- \frac{1}{5} \cdot \ln (x) + \ln (\sqrt{x} \sqrt[4]{x})$

Step 2

Simplify each term.

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Step 2.1

Multiply $- \frac{1}{5} \ln (x)$ .

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Step 2.1.1

Reorder $\ln (x)$ and $\frac{1}{5}$ .

$- (\frac{1}{5} \ln (x)) + \ln (\sqrt{x} \sqrt[4]{x})$

Step 2.1.2

Simplify $\frac{1}{5} \ln (x)$ by moving $\frac{1}{5}$ inside the logarithm.

$- \ln (x^{\frac{1}{5}}) + \ln (\sqrt{x} \sqrt[4]{x})$

Step 2.2

Multiply $\sqrt{x} \sqrt[4]{x}$ .

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Step 2.2.1

Rewrite the expression using the least common index of $4$ .

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Step 2.2.1.1

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

$- \ln (x^{\frac{1}{5}}) + \ln (x^{\frac{1}{2}} \sqrt[4]{x})$

Step 2.2.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{2}{4}}$ .

$- \ln (x^{\frac{1}{5}}) + \ln (x^{\frac{2}{4}} \sqrt[4]{x})$

Step 2.2.1.3

Rewrite $x^{\frac{2}{4}}$ as $\sqrt[4]{x^{2}}$ .

$- \ln (x^{\frac{1}{5}}) + \ln (\sqrt[4]{x^{2}} \sqrt[4]{x})$

Step 2.2.2

Combine using the product rule for radicals.

$- \ln (x^{\frac{1}{5}}) + \ln (\sqrt[4]{x^{2} x})$

Step 2.2.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.2.3.1

Multiply $x^{2}$ by $x$ .

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Step 2.2.3.1.1

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

$- \ln (x^{\frac{1}{5}}) + \ln (\sqrt[4]{x^{2} x^{1}})$

Step 2.2.3.1.2

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

$- \ln (x^{\frac{1}{5}}) + \ln (\sqrt[4]{x^{2 + 1}})$

Step 2.2.3.2

Add $2$ and $1$ .

$- \ln (x^{\frac{1}{5}}) + \ln (\sqrt[4]{x^{3}})$