Precalculus Examples

$\log (\frac{10 x^{4} \sqrt[3]{7 - x}}{3 {(x + 8)}^{2}})$

Step 1

Rewrite $\log (\frac{10 x^{4} \sqrt[3]{7 - x}}{3 {(x + 8)}^{2}})$ as $\log (10 x^{4} \sqrt[3]{7 - x}) - \log (3 {(x + 8)}^{2})$ .

$\log (10 x^{4} \sqrt[3]{7 - x}) - \log (3 {(x + 8)}^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7 - x}$ as ${(7 - x)}^{\frac{1}{3}}$ .

$\log (10 x^{4} {(7 - x)}^{\frac{1}{3}}) - \log (3 {(x + 8)}^{2})$

Step 3

Rewrite $\log (10 x^{4} {(7 - x)}^{\frac{1}{3}})$ as $\log (10 x^{4}) + \log ({(7 - x)}^{\frac{1}{3}})$ .

$\log (10 x^{4}) + \log ({(7 - x)}^{\frac{1}{3}}) - \log (3 {(x + 8)}^{2})$

Step 4

Rewrite $\log (10 x^{4})$ as $\log (10) + \log (x^{4})$ .

$\log (10) + \log (x^{4}) + \log ({(7 - x)}^{\frac{1}{3}}) - \log (3 {(x + 8)}^{2})$

Step 5

Expand $\log (x^{4})$ by moving $4$ outside the logarithm.

$\log (10) + 4 \log (x) + \log ({(7 - x)}^{\frac{1}{3}}) - \log (3 {(x + 8)}^{2})$

Step 6

Expand $\log ({(7 - x)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log (10) + 4 \log (x) + \frac{1}{3} \log (7 - x) - \log (3 {(x + 8)}^{2})$

Step 7

Logarithm base $10$ of $10$ is $1$ .

$1 + 4 \log (x) + \frac{1}{3} \log (7 - x) - \log (3 {(x + 8)}^{2})$

Step 8

Rewrite $\log (3 {(x + 8)}^{2})$ as $\log (3) + \log ({(x + 8)}^{2})$ .

$1 + 4 \log (x) + \frac{1}{3} \log (7 - x) - (\log (3) + \log ({(x + 8)}^{2}))$

Step 9

Expand $\log ({(x + 8)}^{2})$ by moving $2$ outside the logarithm.

$1 + 4 \log (x) + \frac{1}{3} \log (7 - x) - (\log (3) + 2 \log (x + 8))$

Step 10

Simplify each term.

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Step 10.1

Combine $\frac{1}{3}$ and $\log (7 - x)$ .

$1 + 4 \log (x) + \frac{\log (7 - x)}{3} - (\log (3) + 2 \log (x + 8))$

Step 10.2

Apply the distributive property.

$1 + 4 \log (x) + \frac{\log (7 - x)}{3} - \log (3) - (2 \log (x + 8))$

Step 10.3

Multiply $2$ by $- 1$ .

$1 + 4 \log (x) + \frac{\log (7 - x)}{3} - \log (3) - 2 \log (x + 8)$

Step 11

Write $1$ as a fraction with a common denominator.

$4 \log (x) + \frac{3}{3} + \frac{\log (7 - x)}{3} - \log (3) - 2 \log (x + 8)$

Step 12

Combine the numerators over the common denominator.

$4 \log (x) + \frac{3 + \log (7 - x)}{3} - \log (3) - 2 \log (x + 8)$

Step 13

To write $- 2 \log (x + 8)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 \log (x) + \frac{3 + \log (7 - x)}{3} - 2 \log (x + 8) \cdot \frac{3}{3} - \log (3)$

Step 14

Combine $- 2 \log (x + 8)$ and $\frac{3}{3}$ .

$4 \log (x) + \frac{3 + \log (7 - x)}{3} + \frac{- 2 \log (x + 8) \cdot 3}{3} - \log (3)$

Step 15

Combine the numerators over the common denominator.

$4 \log (x) + \frac{3 + \log (7 - x) - 2 \log (x + 8) \cdot 3}{3} - \log (3)$

Step 16

Multiply $3$ by $- 2$ .

$4 \log (x) + \frac{3 + \log (7 - x) - 6 \log (x + 8)}{3} - \log (3)$