Precalculus Examples

$4 \log_{4} (2 + \frac{1}{7}) \log_{4} (r - 7) - \frac{1}{2} \cdot \log_{4} (r)$

Step 1

To write $2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$4 \log_{4} (2 \cdot \frac{7}{7} + \frac{1}{7}) \log_{4} (r - 7) - \frac{1}{2} \cdot \log_{4} (r)$

Step 2

Combine $2$ and $\frac{7}{7}$ .

$4 \log_{4} (\frac{2 \cdot 7}{7} + \frac{1}{7}) \log_{4} (r - 7) - \frac{1}{2} \cdot \log_{4} (r)$

Step 3

Combine the numerators over the common denominator.

$4 \log_{4} (\frac{2 \cdot 7 + 1}{7}) \log_{4} (r - 7) - \frac{1}{2} \cdot \log_{4} (r)$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $2$ by $7$ .

$4 \log_{4} (\frac{14 + 1}{7}) \log_{4} (r - 7) - \frac{1}{2} \cdot \log_{4} (r)$

Step 4.2

Add $14$ and $1$ .

$4 \log_{4} (\frac{15}{7}) \log_{4} (r - 7) - \frac{1}{2} \cdot \log_{4} (r)$

Step 5

Simplify $4 \log_{4} (\frac{15}{7})$ by moving $4$ inside the logarithm.

$\log_{4} ({(\frac{15}{7})}^{4}) \log_{4} (r - 7) - \frac{1}{2} \cdot \log_{4} (r)$

Step 6

Apply the product rule to $\frac{15}{7}$ .

$\log_{4} (\frac{15^{4}}{7^{4}}) \log_{4} (r - 7) - \frac{1}{2} \cdot \log_{4} (r)$

Step 7

Raise $15$ to the power of $4$ .

$\log_{4} (\frac{50625}{7^{4}}) \log_{4} (r - 7) - \frac{1}{2} \cdot \log_{4} (r)$

Step 8

Raise $7$ to the power of $4$ .

$\log_{4} (\frac{50625}{2401}) \log_{4} (r - 7) - \frac{1}{2} \cdot \log_{4} (r)$

Step 9

Multiply $- \frac{1}{2} \log_{4} (r)$ .

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Step 9.1

Reorder $\log_{4} (r)$ and $\frac{1}{2}$ .

$\log_{4} (\frac{50625}{2401}) \log_{4} (r - 7) - (\frac{1}{2} \log_{4} (r))$

Step 9.2

Simplify $\frac{1}{2} \log_{4} (r)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{4} (\frac{50625}{2401}) \log_{4} (r - 7) - \log_{4} (r^{\frac{1}{2}})$