Precalculus Examples

$5 \log_{4} (2) + 7 \log_{4} (x - 4 \log_{4} (y))$

Step 1

Simplify each term.

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Step 1.1

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

$5 (\frac{1}{2}) + 7 \log_{4} (x - 4 \log_{4} (y))$

Step 1.2

Combine $5$ and $\frac{1}{2}$ .

$\frac{5}{2} + 7 \log_{4} (x - 4 \log_{4} (y))$

Step 1.3

Simplify $- 4 \log_{4} (y)$ by moving $4$ inside the logarithm.

$\frac{5}{2} + 7 \log_{4} (x - \log_{4} (y^{4}))$

Step 1.4

Simplify $7 \log_{4} (x - \log_{4} (y^{4}))$ by moving $7$ inside the logarithm.

$\frac{5}{2} + \log_{4} ({(x - \log_{4} (y^{4}))}^{7})$

Step 2

To write $\log_{4} ({(x - \log_{4} (y^{4}))}^{7})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{2} + \log_{4} ({(x - \log_{4} (y^{4}))}^{7}) \cdot \frac{2}{2}$

Step 3

Simplify terms.

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Step 3.1

Combine $\log_{4} ({(x - \log_{4} (y^{4}))}^{7})$ and $\frac{2}{2}$ .

$\frac{5}{2} + \frac{\log_{4} ({(x - \log_{4} (y^{4}))}^{7}) \cdot 2}{2}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{5 + \log_{4} ({(x - \log_{4} (y^{4}))}^{7}) \cdot 2}{2}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $\log_{4} ({(x - \log_{4} (y^{4}))}^{7}) \cdot 2$ .

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Step 4.1.1

Reorder $\log_{4} ({(x - \log_{4} (y^{4}))}^{7})$ and $2$ .

$\frac{5 + 2 \cdot \log_{4} ({(x - \log_{4} (y^{4}))}^{7})}{2}$

Step 4.1.2

Simplify $2 \cdot \log_{4} ({(x - \log_{4} (y^{4}))}^{7})$ by moving $2$ inside the logarithm.

$\frac{5 + \log_{4} ({({(x - \log_{4} (y^{4}))}^{7})}^{2})}{2}$

Step 4.2

Multiply the exponents in ${({(x - \log_{4} (y^{4}))}^{7})}^{2}$ .

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Step 4.2.1

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

$\frac{5 + \log_{4} ({(x - \log_{4} (y^{4}))}^{7 \cdot 2})}{2}$

Step 4.2.2

Multiply $7$ by $2$ .

$\frac{5 + \log_{4} ({(x - \log_{4} (y^{4}))}^{14})}{2}$