Precalculus Examples

$\log (\frac{3 x^{4} \sqrt{y}}{\cos (x) {(x - 1)}^{2}})$

Step 1

Rewrite $\log (\frac{3 x^{4} \sqrt{y}}{\cos (x) {(x - 1)}^{2}})$ as $\log (3 x^{4} \sqrt{y}) - \log (\cos (x) {(x - 1)}^{2})$ .

$\log (3 x^{4} \sqrt{y}) - \log (\cos (x) {(x - 1)}^{2})$

Step 2

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

$\log (3 x^{4} y^{\frac{1}{2}}) - \log (\cos (x) {(x - 1)}^{2})$

Step 3

Rewrite $\log (3 x^{4} y^{\frac{1}{2}})$ as $\log (3 x^{4}) + \log (y^{\frac{1}{2}})$ .

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

Step 4

Rewrite $\log (3 x^{4})$ as $\log (3) + \log (x^{4})$ .

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

Step 5

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

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

Step 6

Expand $\log (y^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 7

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

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

Step 8

Expand $\log ({(x - 1)}^{2})$ by moving $2$ outside the logarithm.

$\log (3) + 4 \log (x) + \frac{1}{2} \log (y) - (\log (\cos (x)) + 2 \log (x - 1))$

Step 9

Simplify each term.

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Step 9.1

Combine $\frac{1}{2}$ and $\log (y)$ .

$\log (3) + 4 \log (x) + \frac{\log (y)}{2} - (\log (\cos (x)) + 2 \log (x - 1))$

Step 9.2

Apply the distributive property.

$\log (3) + 4 \log (x) + \frac{\log (y)}{2} - \log (\cos (x)) - (2 \log (x - 1))$

Step 9.3

Multiply $2$ by $- 1$ .

$\log (3) + 4 \log (x) + \frac{\log (y)}{2} - \log (\cos (x)) - 2 \log (x - 1)$

Step 10

To write $- 2 \log (x - 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log (3) + 4 \log (x) + \frac{\log (y)}{2} - 2 \log (x - 1) \cdot \frac{2}{2} - \log (\cos (x))$

Step 11

Combine $- 2 \log (x - 1)$ and $\frac{2}{2}$ .

$\log (3) + 4 \log (x) + \frac{\log (y)}{2} + \frac{- 2 \log (x - 1) \cdot 2}{2} - \log (\cos (x))$

Step 12

Combine the numerators over the common denominator.

$\log (3) + 4 \log (x) + \frac{\log (y) - 2 \log (x - 1) \cdot 2}{2} - \log (\cos (x))$

Step 13

Multiply $2$ by $- 2$ .

$\log (3) + 4 \log (x) + \frac{\log (y) - 4 \log (x - 1)}{2} - \log (\cos (x))$