Precalculus Examples

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

Step 1

Rewrite $\log_{3} (\frac{4 {(x - 5)}^{2}}{x^{4} {(x - 1)}^{3}})$ as $\log_{3} (4 {(x - 5)}^{2}) - \log_{3} (x^{4} {(x - 1)}^{3})$ .

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

Step 2

Rewrite $\log_{3} (4 {(x - 5)}^{2})$ as $\log_{3} (4) + \log_{3} ({(x - 5)}^{2})$ .

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

Step 3

Expand $\log_{3} ({(x - 5)}^{2})$ by moving $2$ outside the logarithm.

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

Step 4

Rewrite $\log_{3} (x^{4} {(x - 1)}^{3})$ as $\log_{3} (x^{4}) + \log_{3} ({(x - 1)}^{3})$ .

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

Step 5

Expand $\log_{3} (x^{4})$ by moving $4$ outside the logarithm.

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

Step 6

Expand $\log_{3} ({(x - 1)}^{3})$ by moving $3$ outside the logarithm.

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

Step 7

Simplify each term.

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Step 7.1

Rewrite $\log_{3} (4)$ as $\log_{3} (2^{2})$ .

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

Step 7.2

Expand $\log_{3} (2^{2})$ by moving $2$ outside the logarithm.

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

Step 7.3

Apply the distributive property.

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

Step 7.4

Multiply $4$ by $- 1$ .

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

Step 7.5

Multiply $3$ by $- 1$ .

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