Trigonometry Examples

$\log (\sqrt{\frac{a}{b}})$

Step 1

Rewrite $\sqrt{\frac{a}{b}}$ as $\frac{\sqrt{a}}{\sqrt{b}}$ .

$\log (\frac{\sqrt{a}}{\sqrt{b}})$

Step 2

Multiply $\frac{\sqrt{a}}{\sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\log (\frac{\sqrt{a}}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}})$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt{a}}{\sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\log (\frac{\sqrt{a} \sqrt{b}}{\sqrt{b} \sqrt{b}})$

Step 3.2

Raise $\sqrt{b}$ to the power of $1$ .

$\log (\frac{\sqrt{a} \sqrt{b}}{{\sqrt{b}}^{1} \sqrt{b}})$

Step 3.3

Raise $\sqrt{b}$ to the power of $1$ .

$\log (\frac{\sqrt{a} \sqrt{b}}{{\sqrt{b}}^{1} {\sqrt{b}}^{1}})$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (\frac{\sqrt{a} \sqrt{b}}{{\sqrt{b}}^{1 + 1}})$

Step 3.5

Add $1$ and $1$ .

$\log (\frac{\sqrt{a} \sqrt{b}}{{\sqrt{b}}^{2}})$

Step 3.6

Rewrite ${\sqrt{b}}^{2}$ as $b$ .

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Step 3.6.1

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

$\log (\frac{\sqrt{a} \sqrt{b}}{{(b^{\frac{1}{2}})}^{2}})$

Step 3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (\frac{\sqrt{a} \sqrt{b}}{b^{\frac{1}{2} \cdot 2}})$

Step 3.6.3

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

$\log (\frac{\sqrt{a} \sqrt{b}}{b^{\frac{2}{2}}})$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\log (\frac{\sqrt{a} \sqrt{b}}{b^{\frac{2}{2}}})$

Step 3.6.4.2

Rewrite the expression.

$\log (\frac{\sqrt{a} \sqrt{b}}{b^{1}})$

Step 3.6.5

Simplify.

$\log (\frac{\sqrt{a} \sqrt{b}}{b})$

Step 4

Combine using the product rule for radicals.

$\log (\frac{\sqrt{a b}}{b})$