Precalculus Examples

$\log (\sqrt{\frac{x^{9} + 7}{(x^{8} + 5) {(x^{5} - 10)}^{4}}})$

Step 1

Rewrite $\frac{x^{9} + 7}{(x^{8} + 5) {(x^{5} - 10)}^{4}}$ as ${(\frac{1}{{(x^{5} - 10)}^{2}})}^{2} \frac{x^{9} + 7}{{(x^{4})}^{2} + 5}$ .

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Step 1.1

Factor the perfect power $1^{2}$ out of $x^{9} + 7$ .

$\log (\sqrt{\frac{1^{2} (x^{9} + 7)}{(x^{8} + 5) {(x^{5} - 10)}^{4}}})$

Step 1.2

Factor the perfect power ${({(x^{5} - 10)}^{2})}^{2}$ out of $(x^{8} + 5) {(x^{5} - 10)}^{4}$ .

$\log (\sqrt{\frac{1^{2} (x^{9} + 7)}{{({(x^{5} - 10)}^{2})}^{2} ({(x^{4})}^{2} + 5)}})$

Step 1.3

Rearrange the fraction $\frac{1^{2} (x^{9} + 7)}{{({(x^{5} - 10)}^{2})}^{2} ({(x^{4})}^{2} + 5)}$ .

$\log (\sqrt{{(\frac{1}{{(x^{5} - 10)}^{2}})}^{2} \frac{x^{9} + 7}{{(x^{4})}^{2} + 5}})$

Step 2

Pull terms out from under the radical.

$\log (\frac{1}{{(x^{5} - 10)}^{2}} \sqrt{\frac{x^{9} + 7}{{(x^{4})}^{2} + 5}})$

Step 3

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 3.1

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

$\log (\frac{1}{{(x^{5} - 10)}^{2}} \sqrt{\frac{x^{9} + 7}{x^{4 \cdot 2} + 5}})$

Step 3.2

Multiply $4$ by $2$ .

$\log (\frac{1}{{(x^{5} - 10)}^{2}} \sqrt{\frac{x^{9} + 7}{x^{8} + 5}})$

Step 4

Rewrite $\sqrt{\frac{x^{9} + 7}{x^{8} + 5}}$ as $\frac{\sqrt{x^{9} + 7}}{\sqrt{x^{8} + 5}}$ .

$\log (\frac{1}{{(x^{5} - 10)}^{2}} \cdot \frac{\sqrt{x^{9} + 7}}{\sqrt{x^{8} + 5}})$

Step 5

Combine.

$\log (\frac{1 \sqrt{x^{9} + 7}}{{(x^{5} - 10)}^{2} \sqrt{x^{8} + 5}})$

Step 6

Multiply $\sqrt{x^{9} + 7}$ by $1$ .

$\log (\frac{\sqrt{x^{9} + 7}}{{(x^{5} - 10)}^{2} \sqrt{x^{8} + 5}})$

Step 7

Multiply $\frac{\sqrt{x^{9} + 7}}{{(x^{5} - 10)}^{2} \sqrt{x^{8} + 5}}$ by $\frac{\sqrt{x^{8} + 5}}{\sqrt{x^{8} + 5}}$ .

$\log (\frac{\sqrt{x^{9} + 7}}{{(x^{5} - 10)}^{2} \sqrt{x^{8} + 5}} \cdot \frac{\sqrt{x^{8} + 5}}{\sqrt{x^{8} + 5}})$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{\sqrt{x^{9} + 7}}{{(x^{5} - 10)}^{2} \sqrt{x^{8} + 5}}$ by $\frac{\sqrt{x^{8} + 5}}{\sqrt{x^{8} + 5}}$ .

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} \sqrt{x^{8} + 5} \sqrt{x^{8} + 5}})$

Step 8.2

Move $\sqrt{x^{8} + 5}$ .

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} (\sqrt{x^{8} + 5} \sqrt{x^{8} + 5})})$

Step 8.3

Raise $\sqrt{x^{8} + 5}$ to the power of $1$ .

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} ({\sqrt{x^{8} + 5}}^{1} \sqrt{x^{8} + 5})})$

Step 8.4

Raise $\sqrt{x^{8} + 5}$ to the power of $1$ .

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} ({\sqrt{x^{8} + 5}}^{1} {\sqrt{x^{8} + 5}}^{1})})$

Step 8.5

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

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} {\sqrt{x^{8} + 5}}^{1 + 1}})$

Step 8.6

Add $1$ and $1$ .

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} {\sqrt{x^{8} + 5}}^{2}})$

Step 8.7

Rewrite ${\sqrt{x^{8} + 5}}^{2}$ as $x^{8} + 5$ .

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Step 8.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{8} + 5}$ as ${(x^{8} + 5)}^{\frac{1}{2}}$ .

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} {({(x^{8} + 5)}^{\frac{1}{2}})}^{2}})$

Step 8.7.2

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

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} {(x^{8} + 5)}^{\frac{1}{2} \cdot 2}})$

Step 8.7.3

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

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} {(x^{8} + 5)}^{\frac{2}{2}}})$

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} {(x^{8} + 5)}^{\frac{2}{2}}})$

Step 8.7.4.2

Rewrite the expression.

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} {(x^{8} + 5)}^{1}})$

Step 8.7.5

Simplify.

$\log (\frac{\sqrt{x^{9} + 7} \sqrt{x^{8} + 5}}{{(x^{5} - 10)}^{2} (x^{8} + 5)})$

Step 9

Combine using the product rule for radicals.

$\log (\frac{\sqrt{(x^{9} + 7) (x^{8} + 5)}}{{(x^{5} - 10)}^{2} (x^{8} + 5)})$