Algebra Examples

$\log (x \sqrt{\frac{\sqrt{x}}{z}})$

Step 1

Rewrite $\log (x \sqrt{\frac{\sqrt{x}}{z}})$ as $\log (x) + \log (\sqrt{\frac{\sqrt{x}}{z}})$ .

$\log (x) + \log (\sqrt{\frac{\sqrt{x}}{z}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\log (x) + \log (\sqrt{\frac{x^{\frac{1}{2}}}{z}})$

Step 3

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

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

Step 4

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

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

Step 5

Rewrite $\log (\frac{x^{\frac{1}{2}}}{z})$ as $\log (x^{\frac{1}{2}}) - \log (z)$ .

$\log (x) + \frac{1}{2} (\log (x^{\frac{1}{2}}) - \log (z))$

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

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

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

Step 7.2

Apply the distributive property.

$\log (x) + \frac{1}{2} \cdot \frac{\log (x)}{2} + \frac{1}{2} (- \log (z))$

Step 7.3

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

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Step 7.3.1

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

$\log (x) + \frac{\log (x)}{2 \cdot 2} + \frac{1}{2} (- \log (z))$

Step 7.3.2

Multiply $2$ by $2$ .

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

Step 7.4

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

$\log (x) + \frac{\log (x)}{4} - \frac{\log (z)}{2}$

Step 8

To write $\log (x)$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\log (x) \cdot \frac{4}{4} + \frac{\log (x)}{4} - \frac{\log (z)}{2}$

Step 9

Combine $\log (x)$ and $\frac{4}{4}$ .

$\frac{\log (x) \cdot 4}{4} + \frac{\log (x)}{4} - \frac{\log (z)}{2}$

Step 10

Combine the numerators over the common denominator.

$\frac{\log (x) \cdot 4 + \log (x)}{4} - \frac{\log (z)}{2}$

Step 11

Add $\log (x) \cdot 4$ and $\log (x)$ .

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Step 11.1

Reorder $\log (x)$ and $4$ .

$\frac{4 \cdot \log (x) + \log (x)}{4} - \frac{\log (z)}{2}$

Step 11.2

Add $4 \cdot \log (x)$ and $\log (x)$ .

$\frac{5 \cdot \log (x)}{4} - \frac{\log (z)}{2}$

$\frac{5 \log (x)}{4} - \frac{\log (z)}{2}$