Precalculus Examples

$\log (\sqrt{z^{3} x^{4} y})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z^{3} x^{4} y}$ as ${(z^{3} x^{4} y)}^{\frac{1}{2}}$ .

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

Step 2

Expand $\log ({(z^{3} x^{4} y)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Apply the distributive property.

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

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{2} (3 \log (z))$ .

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Step 8.1.1

Combine $3$ and $\frac{1}{2}$ .

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

Step 8.1.2

Combine $\frac{3}{2}$ and $\log (z)$ .

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

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

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

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

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

Step 8.3

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

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