Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log (\sqrt{x})$ by moving $2$ inside the logarithm.

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

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

Step 1.2.5

Simplify.

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

Step 1.3

Simplify $3 \log (x^{\frac{1}{3}})$ by moving $3$ inside the logarithm.

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

Step 1.4

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.4.1

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

$\log (x) + \log (x^{\frac{1}{3} \cdot 3})$

Step 1.4.2

Cancel the common factor of $3$ .

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Step 1.4.2.1

Cancel the common factor.

$\log (x) + \log (x^{\frac{1}{3} \cdot 3})$

Step 1.4.2.2

Rewrite the expression.

$\log (x) + \log (x^{1})$

Step 1.5

Simplify.

$\log (x) + \log (x)$

Step 2

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

$\log (x \cdot x)$

Step 3

Multiply $x$ by $x$ .

$\log (x^{2})$