Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Simplify $2 \log (b)$ by moving $2$ inside the logarithm.

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

Step 1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 2.2

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

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

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Add $1$ and $1$ .

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

Step 2.3.6

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

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Step 2.3.6.1

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

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

Step 2.3.6.2

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

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

Step 2.3.6.3

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

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

Step 2.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.6.4.1

Cancel the common factor.

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

Step 2.3.6.4.2

Rewrite the expression.

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

Step 2.3.6.5

Simplify.

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

Step 2.4

Cancel the common factor of $a$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Divide $b \sqrt{a}$ by $1$ .

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