Algebra Examples

$\log_{3} (\frac{\sqrt{5}}{q^{2} p})$

Step 1

Rewrite $\log_{3} (\frac{\sqrt{5}}{q^{2} p})$ as $\log_{3} (\sqrt{5}) - \log_{3} (q^{2} p)$ .

$\log_{3} (\sqrt{5}) - \log_{3} (q^{2} p)$

Step 2

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

$\log_{3} (5^{\frac{1}{2}}) - \log_{3} (q^{2} p)$

Step 3

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

$\frac{1}{2} \log_{3} (5) - \log_{3} (q^{2} p)$

Step 4

Combine $\frac{1}{2}$ and $\log_{3} (5)$ .

$\frac{\log_{3} (5)}{2} - \log_{3} (q^{2} p)$

Step 5

Rewrite $\log_{3} (q^{2} p)$ as $\log_{3} (q^{2}) + \log_{3} (p)$ .

$\frac{\log_{3} (5)}{2} - (\log_{3} (q^{2}) + \log_{3} (p))$

Step 6

Expand $\log_{3} (q^{2})$ by moving $2$ outside the logarithm.

$\frac{\log_{3} (5)}{2} - (2 \log_{3} (q) + \log_{3} (p))$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{\log_{3} (5)}{2} - (2 \log_{3} (q)) - \log_{3} (p)$

Step 7.2

Multiply $2$ by $- 1$ .

$\frac{\log_{3} (5)}{2} - 2 \log_{3} (q) - \log_{3} (p)$