Algebra Examples

$\log (\frac{\sqrt{9 a^{16}}}{b^{2}})$

Step 1

Rewrite $\log (\frac{\sqrt{9 a^{16}}}{b^{2}})$ as $\log (\sqrt{9 a^{16}}) - \log (b^{2})$ .

$\log (\sqrt{9 a^{16}}) - \log (b^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 a^{16}}$ as ${(9 a^{16})}^{\frac{1}{2}}$ .

$\log ({(9 a^{16})}^{\frac{1}{2}}) - \log (b^{2})$

Step 3

Expand $\log ({(9 a^{16})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (9 a^{16}) - \log (b^{2})$

Step 4

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

$\frac{1}{2} \log (9 a^{16}) - (2 \log (b))$

Step 5

Multiply $2$ by $- 1$ .

$\frac{1}{2} \log (9 a^{16}) - 2 \log (b)$

Step 6

Rewrite $\log (9 a^{16})$ as $\log (9) + \log (a^{16})$ .

$\frac{1}{2} (\log (9) + \log (a^{16})) - 2 \log (b)$

Step 7

Expand $\log (a^{16})$ by moving $16$ outside the logarithm.

$\frac{1}{2} (\log (9) + 16 \log (a)) - 2 \log (b)$

Step 8

Simplify each term.

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Step 8.1

Simplify each term.

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Step 8.1.1

Rewrite $\log (9)$ as $\log (3^{2})$ .

$\frac{1}{2} (\log (3^{2}) + 16 \log (a)) - 2 \log (b)$

Step 8.1.2

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

$\frac{1}{2} (2 \log (3) + 16 \log (a)) - 2 \log (b)$

Step 8.2

Apply the distributive property.

$\frac{1}{2} (2 \log (3)) + \frac{1}{2} (16 \log (a)) - 2 \log (b)$

Step 8.3

Cancel the common factor of $2$ .

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Step 8.3.1

Factor $2$ out of $2 \log (3)$ .

$\frac{1}{2} (2 (\log (3))) + \frac{1}{2} (16 \log (a)) - 2 \log (b)$

Step 8.3.2

Cancel the common factor.

$\frac{1}{2} (2 \log (3)) + \frac{1}{2} (16 \log (a)) - 2 \log (b)$

Step 8.3.3

Rewrite the expression.

$\log (3) + \frac{1}{2} (16 \log (a)) - 2 \log (b)$

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Factor $2$ out of $16 \log (a)$ .

$\log (3) + \frac{1}{2} (2 (8 \log (a))) - 2 \log (b)$

Step 8.4.2

Cancel the common factor.

$\log (3) + \frac{1}{2} (2 (8 \log (a))) - 2 \log (b)$

Step 8.4.3

Rewrite the expression.

$\log (3) + 8 \log (a) - 2 \log (b)$