Precalculus Examples

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x^{3} - x}})$

Step 1

Simplify the denominator.

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Step 1.1

Factor $x$ out of $x^{3} - x$ .

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Step 1.1.1

Factor $x$ out of $x^{3}$ .

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x \cdot x^{2} - x}})$

Step 1.1.2

Factor $x$ out of $- x$ .

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x \cdot x^{2} + x \cdot - 1}})$

Step 1.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 1$ .

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x (x^{2} - 1)}})$

Step 1.2

Rewrite $1$ as $1^{2}$ .

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x (x^{2} - 1^{2})}})$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x (x + 1) (x - 1)}})$

Step 2

Multiply $\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x (x + 1) (x - 1)}}$ by $\frac{\sqrt{x (x + 1) (x - 1)}}{\sqrt{x (x + 1) (x - 1)}}$ .

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x (x + 1) (x - 1)}} \cdot \frac{\sqrt{x (x + 1) (x - 1)}}{\sqrt{x (x + 1) (x - 1)}})$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x (x + 1) (x - 1)}}$ by $\frac{\sqrt{x (x + 1) (x - 1)}}{\sqrt{x (x + 1) (x - 1)}}$ .

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{\sqrt{x (x + 1) (x - 1)} \sqrt{x (x + 1) (x - 1)}})$

Step 3.2

Raise $\sqrt{x (x + 1) (x - 1)}$ to the power of $1$ .

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{{\sqrt{x (x + 1) (x - 1)}}^{1} \sqrt{x (x + 1) (x - 1)}})$

Step 3.3

Raise $\sqrt{x (x + 1) (x - 1)}$ to the power of $1$ .

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{{\sqrt{x (x + 1) (x - 1)}}^{1} {\sqrt{x (x + 1) (x - 1)}}^{1}})$

Step 3.4

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

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{{\sqrt{x (x + 1) (x - 1)}}^{1 + 1}})$

Step 3.5

Add $1$ and $1$ .

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{{\sqrt{x (x + 1) (x - 1)}}^{2}})$

Step 3.6

Rewrite ${\sqrt{x (x + 1) (x - 1)}}^{2}$ as $x (x + 1) (x - 1)$ .

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Step 3.6.1

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

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{{({(x (x + 1) (x - 1))}^{\frac{1}{2}})}^{2}})$

Step 3.6.2

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

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{{(x (x + 1) (x - 1))}^{\frac{1}{2} \cdot 2}})$

Step 3.6.3

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

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{{(x (x + 1) (x - 1))}^{\frac{2}{2}}})$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{{(x (x + 1) (x - 1))}^{\frac{2}{2}}})$

Step 3.6.4.2

Rewrite the expression.

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{{(x (x + 1) (x - 1))}^{1}})$

Step 3.6.5

Simplify.

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{x (x + 1) (x - 1)})$

Step 4

Factor $x$ out of $x^{2} {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}$ .

$\log_{5} (\frac{x (x {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)})}{x (x + 1) (x - 1)})$

Step 5

Cancel the common factors.

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Step 5.1

Factor $x$ out of $x (x + 1) (x - 1)$ .

$\log_{5} (\frac{x (x {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)})}{x ((x + 1) (x - 1))})$

Step 5.2

Cancel the common factor.

$\log_{5} (\frac{x (x {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)})}{x ((x + 1) (x - 1))})$

Step 5.3

Rewrite the expression.

$\log_{5} (\frac{x {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{(x + 1) (x - 1)})$

Step 6

Rewrite $\log_{5} (\frac{x {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{(x + 1) (x - 1)})$ using the change of base formula.

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Step 6.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 6.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\frac{x {(1 - 5 x)}^{\frac{3}{2}} \sqrt{x (x + 1) (x - 1)}}{(x + 1) (x - 1)})}{\log (5)}$