Precalculus Examples

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

Step 1

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

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

Step 2

Rewrite $\log (4 x^{3})$ as $\log (4) + \log (x^{3})$ .

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

Step 3

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

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

Step 4

Rewrite $\log (y^{2} {(x - 1)}^{5})$ as $\log (y^{2}) + \log ({(x - 1)}^{5})$ .

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

Step 5

Expand $\log (y^{2})$ by moving $2$ outside the logarithm.

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

Step 6

Expand $\log ({(x - 1)}^{5})$ by moving $5$ outside the logarithm.

$\log (4) + 3 \log (x) - (2 \log (y) + 5 \log (x - 1))$

Step 7

Simplify each term.

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Step 7.1

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

$\log (2^{2}) + 3 \log (x) - (2 \log (y) + 5 \log (x - 1))$

Step 7.2

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

$2 \log (2) + 3 \log (x) - (2 \log (y) + 5 \log (x - 1))$

Step 7.3

Apply the distributive property.

$2 \log (2) + 3 \log (x) - (2 \log (y)) - (5 \log (x - 1))$

Step 7.4

Multiply $2$ by $- 1$ .

$2 \log (2) + 3 \log (x) - 2 \log (y) - (5 \log (x - 1))$

Step 7.5

Multiply $5$ by $- 1$ .

$2 \log (2) + 3 \log (x) - 2 \log (y) - 5 \log (x - 1)$