Precalculus Examples

$\log (\frac{6 x^{5} y^{3}}{7 z^{2}})$

Step 1

Rewrite $\log (\frac{6 x^{5} y^{3}}{7 z^{2}})$ as $\log (6 x^{5} y^{3}) - \log (7 z^{2})$ .

$\log (6 x^{5} y^{3}) - \log (7 z^{2})$

Step 2

Rewrite $\log (6 x^{5} y^{3})$ as $\log (6 x^{5}) + \log (y^{3})$ .

$\log (6 x^{5}) + \log (y^{3}) - \log (7 z^{2})$

Step 3

Rewrite $\log (6 x^{5})$ as $\log (6) + \log (x^{5})$ .

$\log (6) + \log (x^{5}) + \log (y^{3}) - \log (7 z^{2})$

Step 4

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

$\log (6) + 5 \log (x) + \log (y^{3}) - \log (7 z^{2})$

Step 5

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

$\log (6) + 5 \log (x) + 3 \log (y) - \log (7 z^{2})$

Step 6

Rewrite $\log (7 z^{2})$ as $\log (7) + \log (z^{2})$ .

$\log (6) + 5 \log (x) + 3 \log (y) - (\log (7) + \log (z^{2}))$

Step 7

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

$\log (6) + 5 \log (x) + 3 \log (y) - (\log (7) + 2 \log (z))$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\log (6)$ as $\log (2 (3))$ .

$\log (2 (3)) + 5 \log (x) + 3 \log (y) - (\log (7) + 2 \log (z))$

Step 8.2

Rewrite $\log (2 (3))$ as $\log (2) + \log (3)$ .

$\log (2) + \log (3) + 5 \log (x) + 3 \log (y) - (\log (7) + 2 \log (z))$

Step 8.3

Apply the distributive property.

$\log (2) + \log (3) + 5 \log (x) + 3 \log (y) - \log (7) - (2 \log (z))$

Step 8.4

Multiply $2$ by $- 1$ .

$\log (2) + \log (3) + 5 \log (x) + 3 \log (y) - \log (7) - 2 \log (z)$