Precalculus Examples

$\log_{4} (x + 1) - 2 \log_{4} (x - 1) - 6 \log_{4} (x)$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{4} (x + 1) - \log_{4} ({(x - 1)}^{2}) - 6 \log_{4} (x)$

Step 1.2

Simplify $- 6 \log_{4} (x)$ by moving $6$ inside the logarithm.

$\log_{4} (x + 1) - \log_{4} ({(x - 1)}^{2}) - \log_{4} (x^{6})$

Step 2

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

$\log_{4} (\frac{x + 1}{{(x - 1)}^{2}}) - \log_{4} (x^{6})$

Step 3

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

$\log_{4} (\frac{\frac{x + 1}{{(x - 1)}^{2}}}{x^{6}})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{4} (\frac{x + 1}{{(x - 1)}^{2}} \cdot \frac{1}{x^{6}})$

Step 5

Multiply $\frac{x + 1}{{(x - 1)}^{2}}$ by $\frac{1}{x^{6}}$ .

$\log_{4} (\frac{x + 1}{{(x - 1)}^{2} x^{6}})$

Step 6

Reorder factors in $\log_{4} (\frac{x + 1}{{(x - 1)}^{2} x^{6}})$ .

$\log_{4} (\frac{x + 1}{x^{6} {(x - 1)}^{2}})$