Precalculus Examples

$2 \log_{7} (3 + \frac{1}{4}) + \frac{1}{4} \cdot \log_{7} (x - 5) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1

Simplify each term.

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Step 1.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 \log_{7} (3 \cdot \frac{4}{4} + \frac{1}{4}) + \frac{1}{4} \cdot \log_{7} (x - 5) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1.2

Combine $3$ and $\frac{4}{4}$ .

$2 \log_{7} (\frac{3 \cdot 4}{4} + \frac{1}{4}) + \frac{1}{4} \cdot \log_{7} (x - 5) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1.3

Combine the numerators over the common denominator.

$2 \log_{7} (\frac{3 \cdot 4 + 1}{4}) + \frac{1}{4} \cdot \log_{7} (x - 5) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Multiply $3$ by $4$ .

$2 \log_{7} (\frac{12 + 1}{4}) + \frac{1}{4} \cdot \log_{7} (x - 5) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1.4.2

Add $12$ and $1$ .

$2 \log_{7} (\frac{13}{4}) + \frac{1}{4} \cdot \log_{7} (x - 5) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1.5

Simplify $2 \log_{7} (\frac{13}{4})$ by moving $2$ inside the logarithm.

$\log_{7} ({(\frac{13}{4})}^{2}) + \frac{1}{4} \cdot \log_{7} (x - 5) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1.6

Apply the product rule to $\frac{13}{4}$ .

$\log_{7} (\frac{13^{2}}{4^{2}}) + \frac{1}{4} \cdot \log_{7} (x - 5) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1.7

Raise $13$ to the power of $2$ .

$\log_{7} (\frac{169}{4^{2}}) + \frac{1}{4} \cdot \log_{7} (x - 5) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1.8

Raise $4$ to the power of $2$ .

$\log_{7} (\frac{169}{16}) + \frac{1}{4} \cdot \log_{7} (x - 5) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1.9

Simplify $\frac{1}{4} \log_{7} (x - 5)$ by moving $\frac{1}{4}$ inside the logarithm.

$\log_{7} (\frac{169}{16}) + \log_{7} ({(x - 5)}^{\frac{1}{4}}) - \frac{1}{2} \cdot \log_{7} (x)$

Step 1.10

Multiply $- \frac{1}{2} \log_{7} (x)$ .

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Step 1.10.1

Reorder $\log_{7} (x)$ and $\frac{1}{2}$ .

$\log_{7} (\frac{169}{16}) + \log_{7} ({(x - 5)}^{\frac{1}{4}}) - (\frac{1}{2} \log_{7} (x))$

Step 1.10.2

Simplify $\frac{1}{2} \log_{7} (x)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{7} (\frac{169}{16}) + \log_{7} ({(x - 5)}^{\frac{1}{4}}) - \log_{7} (x^{\frac{1}{2}})$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{7} (\frac{169}{16} {(x - 5)}^{\frac{1}{4}}) - \log_{7} (x^{\frac{1}{2}})$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{7} (\frac{\frac{169}{16} {(x - 5)}^{\frac{1}{4}}}{x^{\frac{1}{2}}})$

Step 4

Combine $\frac{169}{16}$ and ${(x - 5)}^{\frac{1}{4}}$ .

$\log_{7} (\frac{\frac{169 {(x - 5)}^{\frac{1}{4}}}{16}}{x^{\frac{1}{2}}})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log_{7} (\frac{169 {(x - 5)}^{\frac{1}{4}}}{16} \cdot \frac{1}{x^{\frac{1}{2}}})$

Step 6

Combine.

$\log_{7} (\frac{169 {(x - 5)}^{\frac{1}{4}} \cdot 1}{16 x^{\frac{1}{2}}})$

Step 7

Multiply $169$ by $1$ .

$\log_{7} (\frac{169 {(x - 5)}^{\frac{1}{4}}}{16 x^{\frac{1}{2}}})$