Precalculus Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

Simplify $2 \log_{7} (5)$ by moving $2$ inside the logarithm.

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

Step 2.2

Raise $5$ to the power of $2$ .

$\log_{7} (25) + \log_{7} (\frac{x + 1}{x^{2} - 4})$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Rewrite $4$ as $2^{2}$ .

$\log_{7} (25) + \log_{7} (\frac{x + 1}{x^{2} - 2^{2}})$

Step 2.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\log_{7} (25) + \log_{7} (\frac{x + 1}{(x + 2) (x - 2)})$

Step 3

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

$\log_{7} (25 \frac{x + 1}{(x + 2) (x - 2)})$

Step 4

Combine $25$ and $\frac{x + 1}{(x + 2) (x - 2)}$ .

$\log_{7} (\frac{25 (x + 1)}{(x + 2) (x - 2)})$