Precalculus Examples

$5 \log_{3} (x + 1) + 3 \log_{3} (x - 1) - 0.5 \log_{3} (x^{2} + 4) - 7 \log_{3} (x - 3)$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \log_{3} (x + 1)$ by moving $5$ inside the logarithm.

$\log_{3} ({(x + 1)}^{5}) + 3 \log_{3} (x - 1) - 0.5 \log_{3} (x^{2} + 4) - 7 \log_{3} (x - 3)$

Step 1.2

Simplify $3 \log_{3} (x - 1)$ by moving $3$ inside the logarithm.

$\log_{3} ({(x + 1)}^{5}) + \log_{3} ({(x - 1)}^{3}) - 0.5 \log_{3} (x^{2} + 4) - 7 \log_{3} (x - 3)$

Step 1.3

Simplify $- 0.5 \log_{3} (x^{2} + 4)$ by moving $0.5$ inside the logarithm.

$\log_{3} ({(x + 1)}^{5}) + \log_{3} ({(x - 1)}^{3}) - \log_{3} ({(x^{2} + 4)}^{0.5}) - 7 \log_{3} (x - 3)$

Step 1.4

Simplify $- 7 \log_{3} (x - 3)$ by moving $7$ inside the logarithm.

$\log_{3} ({(x + 1)}^{5}) + \log_{3} ({(x - 1)}^{3}) - \log_{3} ({(x^{2} + 4)}^{0.5}) - \log_{3} ({(x - 3)}^{7})$

Step 2

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

$\log_{3} ({(x + 1)}^{5} {(x - 1)}^{3}) - \log_{3} ({(x^{2} + 4)}^{0.5}) - \log_{3} ({(x - 3)}^{7})$

Step 3

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

$\log_{3} (\frac{{(x + 1)}^{5} {(x - 1)}^{3}}{{(x^{2} + 4)}^{0.5}}) - \log_{3} ({(x - 3)}^{7})$

Step 4

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

$\log_{3} (\frac{\frac{{(x + 1)}^{5} {(x - 1)}^{3}}{{(x^{2} + 4)}^{0.5}}}{{(x - 3)}^{7}})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log_{3} (\frac{{(x + 1)}^{5} {(x - 1)}^{3}}{{(x^{2} + 4)}^{0.5}} \cdot \frac{1}{{(x - 3)}^{7}})$

Step 6

Combine.

$\log_{3} (\frac{{(x + 1)}^{5} {(x - 1)}^{3} \cdot 1}{{(x^{2} + 4)}^{0.5} {(x - 3)}^{7}})$

Step 7

Multiply ${(x + 1)}^{5}$ by $1$ .

$\log_{3} (\frac{{(x + 1)}^{5} {(x - 1)}^{3}}{{(x^{2} + 4)}^{0.5} {(x - 3)}^{7}})$