Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Apply the distributive property.

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

Step 8

Simplify.

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Step 8.1

Cancel the common factor of $2$ .

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Step 8.1.1

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

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

Step 8.1.2

Cancel the common factor.

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

Step 8.1.3

Rewrite the expression.

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.3

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

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