Algebra Examples

$\ln (x^{6}) - 2 \ln (\sqrt{x^{5}})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $x^{5}$ as ${(x^{2})}^{2} x$ .

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Step 1.1.1

Factor out $x^{4}$ .

$\ln (x^{6}) - 2 \ln (\sqrt{x^{4} x})$

Step 1.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

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

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

Step 1.4

Apply the product rule to $x^{2} \sqrt{x}$ .

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

Multiply $2$ by $2$ .

$\ln (x^{6}) - \ln (x^{4} {\sqrt{x}}^{2})$

Step 1.6

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

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Step 1.6.1

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

$\ln (x^{6}) - \ln (x^{4} {(x^{\frac{1}{2}})}^{2})$

Step 1.6.2

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

$\ln (x^{6}) - \ln (x^{4} x^{\frac{1}{2} \cdot 2})$

Step 1.6.3

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

$\ln (x^{6}) - \ln (x^{4} x^{\frac{2}{2}})$

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

$\ln (x^{6}) - \ln (x^{4} x^{\frac{2}{2}})$

Step 1.6.4.2

Rewrite the expression.

$\ln (x^{6}) - \ln (x^{4} x^{1})$

Step 1.6.5

Simplify.

$\ln (x^{6}) - \ln (x^{4} x)$

Step 1.7

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 1.7.1

Multiply $x^{4}$ by $x$ .

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Step 1.7.1.1

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

$\ln (x^{6}) - \ln (x^{4} x^{1})$

Step 1.7.1.2

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

$\ln (x^{6}) - \ln (x^{4 + 1})$

Step 1.7.2

Add $4$ and $1$ .

$\ln (x^{6}) - \ln (x^{5})$

Step 2

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

$\ln (\frac{x^{6}}{x^{5}})$

Step 3

Cancel the common factor of $x^{6}$ and $x^{5}$ .

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Step 3.1

Factor $x^{5}$ out of $x^{6}$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.2.4

Divide $x$ by $1$ .

$\ln (x)$