Precalculus Examples

$\log_{4} (\frac{8 m}{\sqrt{2} n})$

Step 1

Rewrite $\log_{4} (\frac{8 m}{\sqrt{2} n})$ as $\log_{4} (8 m) - \log_{4} (\sqrt{2} n)$ .

$\log_{4} (8 m) - \log_{4} (\sqrt{2} n)$

Step 2

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

$\log_{4} (8 m) - \log_{4} (2^{\frac{1}{2}} n)$

Step 3

Rewrite $\log_{4} (8 m)$ as $\log_{4} (8) + \log_{4} (m)$ .

$\log_{4} (8) + \log_{4} (m) - \log_{4} (2^{\frac{1}{2}} n)$

Step 4

Logarithm base $4$ of $8$ is $\frac{3}{2}$ .

$\frac{3}{2} + \log_{4} (m) - \log_{4} (2^{\frac{1}{2}} n)$

Step 5

Rewrite $\log_{4} (2^{\frac{1}{2}} n)$ as $\log_{4} (2^{\frac{1}{2}}) + \log_{4} (n)$ .

$\frac{3}{2} + \log_{4} (m) - (\log_{4} (2^{\frac{1}{2}}) + \log_{4} (n))$

Step 6

Expand $\log_{4} (2^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{3}{2} + \log_{4} (m) - (\frac{1}{2} \log_{4} (2) + \log_{4} (n))$

Step 7

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$\frac{3}{2} + \log_{4} (m) - (\frac{1}{2} \cdot \frac{1}{2} + \log_{4} (n))$

Step 8

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 8.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{3}{2} + \log_{4} (m) - (\frac{1}{2 \cdot 2} + \log_{4} (n))$

Step 8.2

Multiply $2$ by $2$ .

$\frac{3}{2} + \log_{4} (m) - (\frac{1}{4} + \log_{4} (n))$

Step 9

Apply the distributive property.

$\frac{3}{2} + \log_{4} (m) - \frac{1}{4} - \log_{4} (n)$

Step 10

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} \cdot \frac{2}{2} - \frac{1}{4} + \log_{4} (m) - \log_{4} (n)$

Step 11

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.1

Multiply $\frac{3}{2}$ by $\frac{2}{2}$ .

$\frac{3 \cdot 2}{2 \cdot 2} - \frac{1}{4} + \log_{4} (m) - \log_{4} (n)$

Step 11.2

Multiply $2$ by $2$ .

$\frac{3 \cdot 2}{4} - \frac{1}{4} + \log_{4} (m) - \log_{4} (n)$

Step 12

Combine the numerators over the common denominator.

$\frac{3 \cdot 2 - 1}{4} + \log_{4} (m) - \log_{4} (n)$

Step 13

Simplify the numerator.

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Step 13.1

Multiply $3$ by $2$ .

$\frac{6 - 1}{4} + \log_{4} (m) - \log_{4} (n)$

Step 13.2

Subtract $1$ from $6$ .

$\frac{5}{4} + \log_{4} (m) - \log_{4} (n)$