Precalculus Examples

$\log (\sqrt{x^{4} y^{- 7}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4} y^{- 7}}$ as ${(x^{4} y^{- 7})}^{\frac{1}{2}}$ .

$\log ({(x^{4} y^{- 7})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(x^{4} y^{- 7})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (x^{4} y^{- 7})$

Step 3

Rewrite $\log (x^{4} y^{- 7})$ as $\log (x^{4}) + \log (y^{- 7})$ .

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

Step 4

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

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

Step 5

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

$\frac{1}{2} (4 \log (x) - 7 \log (y))$

Step 6

Apply the distributive property.

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

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Factor $2$ out of $4 \log (x)$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Multiply $\frac{1}{2} (- 7 \log (y))$ .

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Step 8.1

Combine $- 7$ and $\frac{1}{2}$ .

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

Step 8.2

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

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

Step 9

Move the negative in front of the fraction.

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