Precalculus Examples

$\frac{\log (100 x^{3} \sqrt[3]{5 - x})}{3 {(x + 7)}^{2}}$

Step 1

Rewrite $\log (100 x^{3} \sqrt[3]{5 - x})$ as $\log (100 x^{3}) + \log (\sqrt[3]{5 - x})$ .

$\frac{\log (100 x^{3}) + \log (\sqrt[3]{5 - x})}{3 {(x + 7)}^{2}}$

Step 2

Rewrite $\log (100 x^{3})$ as $\log (100) + \log (x^{3})$ .

$\frac{\log (100) + \log (x^{3}) + \log (\sqrt[3]{5 - x})}{3 {(x + 7)}^{2}}$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 - x}$ as ${(5 - x)}^{\frac{1}{3}}$ .

$\frac{\log (100) + \log (x^{3}) + \log ({(5 - x)}^{\frac{1}{3}})}{3 {(x + 7)}^{2}}$

Step 4

Expand $\log (x^{3})$ by moving $3$ outside the logarithm.

$\frac{\log (100) + 3 \log (x) + \log ({(5 - x)}^{\frac{1}{3}})}{3 {(x + 7)}^{2}}$

Step 5

Expand $\log ({(5 - x)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{\log (100) + 3 \log (x) + \frac{1}{3} \log (5 - x)}{3 {(x + 7)}^{2}}$

Step 6

Logarithm base $10$ of $100$ is $2$ .

$\frac{2 + 3 \log (x) + \frac{1}{3} \log (5 - x)}{3 {(x + 7)}^{2}}$

Step 7

Simplify the numerator.

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Step 7.1

Combine $\frac{1}{3}$ and $\log (5 - x)$ .

$\frac{2 + 3 \log (x) + \frac{\log (5 - x)}{3}}{3 {(x + 7)}^{2}}$

Step 7.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 \log (x) + 2 \cdot \frac{3}{3} + \frac{\log (5 - x)}{3}}{3 {(x + 7)}^{2}}$

Step 7.3

Combine $2$ and $\frac{3}{3}$ .

$\frac{3 \log (x) + \frac{2 \cdot 3}{3} + \frac{\log (5 - x)}{3}}{3 {(x + 7)}^{2}}$

Step 7.4

Combine the numerators over the common denominator.

$\frac{3 \log (x) + \frac{2 \cdot 3 + \log (5 - x)}{3}}{3 {(x + 7)}^{2}}$

Step 7.5

Multiply $2$ by $3$ .

$\frac{3 \log (x) + \frac{6 + \log (5 - x)}{3}}{3 {(x + 7)}^{2}}$

Step 7.6

To write $3 \log (x)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 \log (x) \cdot \frac{3}{3} + \frac{6 + \log (5 - x)}{3}}{3 {(x + 7)}^{2}}$

Step 7.7

Combine $3 \log (x)$ and $\frac{3}{3}$ .

$\frac{\frac{3 \log (x) \cdot 3}{3} + \frac{6 + \log (5 - x)}{3}}{3 {(x + 7)}^{2}}$

Step 7.8

Combine the numerators over the common denominator.

$\frac{\frac{3 \log (x) \cdot 3 + 6 + \log (5 - x)}{3}}{3 {(x + 7)}^{2}}$

Step 7.9

Multiply $3$ by $3$ .

$\frac{\frac{9 \log (x) + 6 + \log (5 - x)}{3}}{3 {(x + 7)}^{2}}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$\frac{9 \log (x) + 6 + \log (5 - x)}{3} \cdot \frac{1}{3 {(x + 7)}^{2}}$

Step 9

Multiply $\frac{9 \log (x) + 6 + \log (5 - x)}{3} \cdot \frac{1}{3 {(x + 7)}^{2}}$ .

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Step 9.1

Multiply $\frac{9 \log (x) + 6 + \log (5 - x)}{3}$ by $\frac{1}{3 {(x + 7)}^{2}}$ .

$\frac{9 \log (x) + 6 + \log (5 - x)}{3 (3 {(x + 7)}^{2})}$

Step 9.2

Multiply $3$ by $3$ .

$\frac{9 \log (x) + 6 + \log (5 - x)}{9 {(x + 7)}^{2}}$