Precalculus Examples

$\log_{c} (\sqrt{x^{3} y^{2} z^{5}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} y^{2} z^{5}}$ as ${(x^{3} y^{2} z^{5})}^{\frac{1}{2}}$ .

$\log_{c} ({(x^{3} y^{2} z^{5})}^{\frac{1}{2}})$

Step 2

Expand $\log_{c} ({(x^{3} y^{2} z^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{c} (x^{3} y^{2} z^{5})$

Step 3

Rewrite $\log_{c} (x^{3} y^{2} z^{5})$ as $\log_{c} (x^{3} y^{2}) + \log_{c} (z^{5})$ .

$\frac{1}{2} (\log_{c} (x^{3} y^{2}) + \log_{c} (z^{5}))$

Step 4

Rewrite $\log_{c} (x^{3} y^{2})$ as $\log_{c} (x^{3}) + \log_{c} (y^{2})$ .

$\frac{1}{2} (\log_{c} (x^{3}) + \log_{c} (y^{2}) + \log_{c} (z^{5}))$

Step 5

Expand $\log_{c} (x^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} (3 \log_{c} (x) + \log_{c} (y^{2}) + \log_{c} (z^{5}))$

Step 6

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

$\frac{1}{2} (3 \log_{c} (x) + 2 \log_{c} (y) + \log_{c} (z^{5}))$

Step 7

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

$\frac{1}{2} (3 \log_{c} (x) + 2 \log_{c} (y) + 5 \log_{c} (z))$

Step 8

Apply the distributive property.

$\frac{1}{2} (3 \log_{c} (x)) + \frac{1}{2} (2 \log_{c} (y)) + \frac{1}{2} (5 \log_{c} (z))$

Step 9

Simplify.

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Step 9.1

Multiply $\frac{1}{2} (3 \log_{c} (x))$ .

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Step 9.1.1

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

$\frac{3}{2} \log_{c} (x) + \frac{1}{2} (2 \log_{c} (y)) + \frac{1}{2} (5 \log_{c} (z))$

Step 9.1.2

Combine $\frac{3}{2}$ and $\log_{c} (x)$ .

$\frac{3 \log_{c} (x)}{2} + \frac{1}{2} (2 \log_{c} (y)) + \frac{1}{2} (5 \log_{c} (z))$

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

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

$\frac{3 \log_{c} (x)}{2} + \frac{1}{2} (2 (\log_{c} (y))) + \frac{1}{2} (5 \log_{c} (z))$

Step 9.2.2

Cancel the common factor.

$\frac{3 \log_{c} (x)}{2} + \frac{1}{2} (2 \log_{c} (y)) + \frac{1}{2} (5 \log_{c} (z))$

Step 9.2.3

Rewrite the expression.

$\frac{3 \log_{c} (x)}{2} + \log_{c} (y) + \frac{1}{2} (5 \log_{c} (z))$

Step 9.3

Multiply $\frac{1}{2} (5 \log_{c} (z))$ .

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Step 9.3.1

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

$\frac{3 \log_{c} (x)}{2} + \log_{c} (y) + \frac{5}{2} \log_{c} (z)$

Step 9.3.2

Combine $\frac{5}{2}$ and $\log_{c} (z)$ .

$\frac{3 \log_{c} (x)}{2} + \log_{c} (y) + \frac{5 \log_{c} (z)}{2}$