Precalculus Examples

$\frac{\log (A^{2} B^{2})}{\sqrt{c}}$

Step 1

Rewrite $\log (A^{2} B^{2})$ as $\log (A^{2}) + \log (B^{2})$ .

$\frac{\log (A^{2}) + \log (B^{2})}{\sqrt{c}}$

Step 2

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

$\frac{2 \log (A) + \log (B^{2})}{\sqrt{c}}$

Step 3

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

$\frac{2 \log (A) + 2 \log (B)}{\sqrt{c}}$

Step 4

Factor $2$ out of $2 \log (A) + 2 \log (B)$ .

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Step 4.1

Factor $2$ out of $2 \log (A)$ .

$\frac{2 (\log (A)) + 2 \log (B)}{\sqrt{c}}$

Step 4.2

Factor $2$ out of $2 \log (B)$ .

$\frac{2 (\log (A)) + 2 (\log (B))}{\sqrt{c}}$

Step 4.3

Factor $2$ out of $2 (\log (A)) + 2 (\log (B))$ .

$\frac{2 (\log (A) + \log (B))}{\sqrt{c}}$

Step 5

Multiply $\frac{2 (\log (A) + \log (B))}{\sqrt{c}}$ by $\frac{\sqrt{c}}{\sqrt{c}}$ .

$\frac{2 (\log (A) + \log (B))}{\sqrt{c}} \cdot \frac{\sqrt{c}}{\sqrt{c}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{2 (\log (A) + \log (B))}{\sqrt{c}}$ by $\frac{\sqrt{c}}{\sqrt{c}}$ .

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{\sqrt{c} \sqrt{c}}$

Step 6.2

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

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{{\sqrt{c}}^{1} \sqrt{c}}$

Step 6.3

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

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{{\sqrt{c}}^{1} {\sqrt{c}}^{1}}$

Step 6.4

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

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{{\sqrt{c}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{{\sqrt{c}}^{2}}$

Step 6.6

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

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Step 6.6.1

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

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{{(c^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{c^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{c^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{c^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{c^{1}}$

Step 6.6.5

Simplify.

$\frac{2 (\log (A) + \log (B)) \sqrt{c}}{c}$

Step 7

Reorder factors in $\frac{2 (\log (A) + \log (B)) \sqrt{c}}{c}$ .

$\frac{2 \sqrt{c} (\log (A) + \log (B))}{c}$