Precalculus Examples

$\log (\frac{\sqrt{x} y^{3}}{x^{7}})$

Step 1

Rewrite $\log (\frac{\sqrt{x} y^{3}}{x^{7}})$ as $\log (\sqrt{x} y^{3}) - \log (x^{7})$ .

$\log (\sqrt{x} y^{3}) - \log (x^{7})$

Step 2

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

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

Step 3

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

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

Step 4

Multiply $7$ by $- 1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

To write $- 7 \log (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 10

Combine $- 7 \log (x)$ and $\frac{2}{2}$ .

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Simplify each term.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Factor $\log (x)$ out of $\log (x) - 7 \log (x) \cdot 2$ .

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Step 12.1.1.1

Multiply by $1$ .

$3 \log (y) + \frac{\log (x) \cdot 1 - 7 \log (x) \cdot 2}{2}$

Step 12.1.1.2

Factor $\log (x)$ out of $- 7 \log (x) \cdot 2$ .

$3 \log (y) + \frac{\log (x) \cdot 1 + \log (x) (- 7 \cdot 2)}{2}$

Step 12.1.1.3

Factor $\log (x)$ out of $\log (x) \cdot 1 + \log (x) (- 7 \cdot 2)$ .

$3 \log (y) + \frac{\log (x) (1 - 7 \cdot 2)}{2}$

Step 12.1.2

Multiply $- 7$ by $2$ .

$3 \log (y) + \frac{\log (x) (1 - 14)}{2}$

Step 12.1.3

Subtract $14$ from $1$ .

$3 \log (y) + \frac{\log (x) \cdot - 13}{2}$

Step 12.2

Move $- 13$ to the left of $\log (x)$ .

$3 \log (y) + \frac{- 13 \cdot \log (x)}{2}$

Step 12.3

Move the negative in front of the fraction.

$3 \log (y) - \frac{13 \log (x)}{2}$