Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \ln (x^{2})$ by moving $3$ inside the logarithm.

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

Step 1.2

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 1.2.1

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

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

Step 1.2.2

Multiply $2$ by $3$ .

$\ln (x^{6}) - \ln (\sqrt{x})$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{x^{6}}{\sqrt{x}})$

Step 3

Multiply $\frac{x^{6}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\ln (\frac{x^{6}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}})$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{x^{6}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\ln (\frac{x^{6} \sqrt{x}}{\sqrt{x} \sqrt{x}})$

Step 4.2

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

$\ln (\frac{x^{6} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}})$

Step 4.3

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

$\ln (\frac{x^{6} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}})$

Step 4.4

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

$\ln (\frac{x^{6} \sqrt{x}}{{\sqrt{x}}^{1 + 1}})$

Step 4.5

Add $1$ and $1$ .

$\ln (\frac{x^{6} \sqrt{x}}{{\sqrt{x}}^{2}})$

Step 4.6

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

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Step 4.6.1

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

$\ln (\frac{x^{6} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}})$

Step 4.6.2

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

$\ln (\frac{x^{6} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}})$

Step 4.6.3

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

$\ln (\frac{x^{6} \sqrt{x}}{x^{\frac{2}{2}}})$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\ln (\frac{x^{6} \sqrt{x}}{x^{\frac{2}{2}}})$

Step 4.6.4.2

Rewrite the expression.

$\ln (\frac{x^{6} \sqrt{x}}{x^{1}})$

Step 4.6.5

Simplify.

$\ln (\frac{x^{6} \sqrt{x}}{x})$

Step 5

Cancel the common factor of $x^{6}$ and $x$ .

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Step 5.1

Factor $x$ out of $x^{6} \sqrt{x}$ .

$\ln (\frac{x (x^{5} \sqrt{x})}{x})$

Step 5.2

Cancel the common factors.

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Step 5.2.1

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

$\ln (\frac{x (x^{5} \sqrt{x})}{x^{1}})$

Step 5.2.2

Factor $x$ out of $x^{1}$ .

$\ln (\frac{x (x^{5} \sqrt{x})}{x \cdot 1})$

Step 5.2.3

Cancel the common factor.

$\ln (\frac{x (x^{5} \sqrt{x})}{x \cdot 1})$

Step 5.2.4

Rewrite the expression.

$\ln (\frac{x^{5} \sqrt{x}}{1})$

Step 5.2.5

Divide $x^{5} \sqrt{x}$ by $1$ .

$\ln (x^{5} \sqrt{x})$