Precalculus Examples

$3 \ln (\sqrt{x}) + \frac{1}{2} \cdot \ln (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \ln (\sqrt{x})$ by moving $3$ inside the logarithm.

$\ln ({\sqrt{x}}^{3}) + \frac{1}{2} \cdot \ln (x)$

Step 1.2

Rewrite ${\sqrt{x}}^{3}$ as $\sqrt{x^{3}}$ .

$\ln (\sqrt{x^{3}}) + \frac{1}{2} \cdot \ln (x)$

Step 1.3

Factor out $x^{2}$ .

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

Step 1.4

Pull terms out from under the radical.

$\ln (x \sqrt{x}) + \frac{1}{2} \cdot \ln (x)$

Step 1.5

Simplify $\frac{1}{2} \ln (x)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 2

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

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

Step 3

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

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Step 3.1

Move $x^{\frac{1}{2}}$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Add $1$ and $2$ .

$\ln (x^{\frac{3}{2}} \sqrt{x})$

Step 4

Multiply $x^{\frac{3}{2}} \sqrt{x}$ .

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Step 4.1

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

$\ln (x^{\frac{3}{2}} x^{\frac{1}{2}})$

Step 4.2

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

$\ln (x^{\frac{3}{2} + \frac{1}{2}})$

Step 4.3

Combine the numerators over the common denominator.

$\ln (x^{\frac{3 + 1}{2}})$

Step 4.4

Add $3$ and $1$ .

$\ln (x^{\frac{4}{2}})$

Step 4.5

Cancel the common factor of $4$ and $2$ .

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Step 4.5.1

Factor $2$ out of $4$ .

$\ln (x^{\frac{2 \cdot 2}{2}})$

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $2$ out of $2$ .

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

Step 4.5.2.2

Cancel the common factor.

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

Step 4.5.2.3

Rewrite the expression.

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

Step 4.5.2.4

Divide $2$ by $1$ .

$\ln (x^{2})$