Precalculus Examples

$\log_{2} (\sqrt{\frac{q^{2} k^{4}}{z w^{5}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{q^{2} k^{4}}{z w^{5}}}$ as ${(\frac{q^{2} k^{4}}{z w^{5}})}^{\frac{1}{2}}$ .

$\log_{2} ({(\frac{q^{2} k^{4}}{z w^{5}})}^{\frac{1}{2}})$

Step 2

Expand $\log_{2} ({(\frac{q^{2} k^{4}}{z w^{5}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{2} (\frac{q^{2} k^{4}}{z w^{5}})$

Step 3

Rewrite $\log_{2} (\frac{q^{2} k^{4}}{z w^{5}})$ as $\log_{2} (q^{2} k^{4}) - \log_{2} (z w^{5})$ .

$\frac{1}{2} (\log_{2} (q^{2} k^{4}) - \log_{2} (z w^{5}))$

Step 4

Rewrite $\log_{2} (q^{2} k^{4})$ as $\log_{2} (q^{2}) + \log_{2} (k^{4})$ .

$\frac{1}{2} (\log_{2} (q^{2}) + \log_{2} (k^{4}) - \log_{2} (z w^{5}))$

Step 5

Expand $\log_{2} (q^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{2} (2 \log_{2} (q) + \log_{2} (k^{4}) - \log_{2} (z w^{5}))$

Step 6

Expand $\log_{2} (k^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{2} (2 \log_{2} (q) + 4 \log_{2} (k) - \log_{2} (z w^{5}))$

Step 7

Rewrite $\log_{2} (z w^{5})$ as $\log_{2} (z) + \log_{2} (w^{5})$ .

$\frac{1}{2} (2 \log_{2} (q) + 4 \log_{2} (k) - (\log_{2} (z) + \log_{2} (w^{5})))$

Step 8

Expand $\log_{2} (w^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{2} (2 \log_{2} (q) + 4 \log_{2} (k) - (\log_{2} (z) + 5 \log_{2} (w)))$

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$\frac{1}{2} (2 \log_{2} (q) + 4 \log_{2} (k) - \log_{2} (z) - (5 \log_{2} (w)))$

Step 9.2

Multiply $5$ by $- 1$ .

$\frac{1}{2} (2 \log_{2} (q) + 4 \log_{2} (k) - \log_{2} (z) - 5 \log_{2} (w))$

Step 10

Apply the distributive property.

$\frac{1}{2} (2 \log_{2} (q)) + \frac{1}{2} (4 \log_{2} (k)) + \frac{1}{2} (- \log_{2} (z)) + \frac{1}{2} (- 5 \log_{2} (w))$

Step 11

Simplify.

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Step 11.1

Cancel the common factor of $2$ .

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Step 11.1.1

Factor $2$ out of $2 \log_{2} (q)$ .

$\frac{1}{2} (2 (\log_{2} (q))) + \frac{1}{2} (4 \log_{2} (k)) + \frac{1}{2} (- \log_{2} (z)) + \frac{1}{2} (- 5 \log_{2} (w))$

Step 11.1.2

Cancel the common factor.

$\frac{1}{2} (2 \log_{2} (q)) + \frac{1}{2} (4 \log_{2} (k)) + \frac{1}{2} (- \log_{2} (z)) + \frac{1}{2} (- 5 \log_{2} (w))$

Step 11.1.3

Rewrite the expression.

$\log_{2} (q) + \frac{1}{2} (4 \log_{2} (k)) + \frac{1}{2} (- \log_{2} (z)) + \frac{1}{2} (- 5 \log_{2} (w))$

Step 11.2

Cancel the common factor of $2$ .

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Step 11.2.1

Factor $2$ out of $4 \log_{2} (k)$ .

$\log_{2} (q) + \frac{1}{2} (2 (2 \log_{2} (k))) + \frac{1}{2} (- \log_{2} (z)) + \frac{1}{2} (- 5 \log_{2} (w))$

Step 11.2.2

Cancel the common factor.

$\log_{2} (q) + \frac{1}{2} (2 (2 \log_{2} (k))) + \frac{1}{2} (- \log_{2} (z)) + \frac{1}{2} (- 5 \log_{2} (w))$

Step 11.2.3

Rewrite the expression.

$\log_{2} (q) + 2 \log_{2} (k) + \frac{1}{2} (- \log_{2} (z)) + \frac{1}{2} (- 5 \log_{2} (w))$

Step 11.3

Combine $\frac{1}{2}$ and $\log_{2} (z)$ .

$\log_{2} (q) + 2 \log_{2} (k) - \frac{\log_{2} (z)}{2} + \frac{1}{2} (- 5 \log_{2} (w))$

Step 11.4

Multiply $\frac{1}{2} (- 5 \log_{2} (w))$ .

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Step 11.4.1

Combine $- 5$ and $\frac{1}{2}$ .

$\log_{2} (q) + 2 \log_{2} (k) - \frac{\log_{2} (z)}{2} + \frac{- 5}{2} \log_{2} (w)$

Step 11.4.2

Combine $\frac{- 5}{2}$ and $\log_{2} (w)$ .

$\log_{2} (q) + 2 \log_{2} (k) - \frac{\log_{2} (z)}{2} + \frac{- 5 \log_{2} (w)}{2}$

Step 12

Move the negative in front of the fraction.

$\log_{2} (q) + 2 \log_{2} (k) - \frac{\log_{2} (z)}{2} - \frac{5 \log_{2} (w)}{2}$