Precalculus Examples

$\log (\frac{x^{2} y^{3} z^{5}}{a^{4} b^{6} c^{7}})$

Step 1

Rewrite $\log (\frac{x^{2} y^{3} z^{5}}{a^{4} b^{6} c^{7}})$ as $\log (x^{2} y^{3} z^{5}) - \log (a^{4} b^{6} c^{7})$ .

$\log (x^{2} y^{3} z^{5}) - \log (a^{4} b^{6} c^{7})$

Step 2

Rewrite $\log (x^{2} y^{3} z^{5})$ as $\log (x^{2} y^{3}) + \log (z^{5})$ .

$\log (x^{2} y^{3}) + \log (z^{5}) - \log (a^{4} b^{6} c^{7})$

Step 3

Rewrite $\log (x^{2} y^{3})$ as $\log (x^{2}) + \log (y^{3})$ .

$\log (x^{2}) + \log (y^{3}) + \log (z^{5}) - \log (a^{4} b^{6} c^{7})$

Step 4

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

$2 \log (x) + \log (y^{3}) + \log (z^{5}) - \log (a^{4} b^{6} c^{7})$

Step 5

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

$2 \log (x) + 3 \log (y) + \log (z^{5}) - \log (a^{4} b^{6} c^{7})$

Step 6

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

$2 \log (x) + 3 \log (y) + 5 \log (z) - \log (a^{4} b^{6} c^{7})$

Step 7

Rewrite $\log (a^{4} b^{6} c^{7})$ as $\log (a^{4} b^{6}) + \log (c^{7})$ .

$2 \log (x) + 3 \log (y) + 5 \log (z) - (\log (a^{4} b^{6}) + \log (c^{7}))$

Step 8

Rewrite $\log (a^{4} b^{6})$ as $\log (a^{4}) + \log (b^{6})$ .

$2 \log (x) + 3 \log (y) + 5 \log (z) - (\log (a^{4}) + \log (b^{6}) + \log (c^{7}))$

Step 9

Expand $\log (a^{4})$ by moving $4$ outside the logarithm.

$2 \log (x) + 3 \log (y) + 5 \log (z) - (4 \log (a) + \log (b^{6}) + \log (c^{7}))$

Step 10

Expand $\log (b^{6})$ by moving $6$ outside the logarithm.

$2 \log (x) + 3 \log (y) + 5 \log (z) - (4 \log (a) + 6 \log (b) + \log (c^{7}))$

Step 11

Expand $\log (c^{7})$ by moving $7$ outside the logarithm.

$2 \log (x) + 3 \log (y) + 5 \log (z) - (4 \log (a) + 6 \log (b) + 7 \log (c))$

Step 12

Simplify each term.

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Step 12.1

Apply the distributive property.

$2 \log (x) + 3 \log (y) + 5 \log (z) - (4 \log (a)) - (6 \log (b)) - (7 \log (c))$

Step 12.2

Simplify.

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Step 12.2.1

Multiply $4$ by $- 1$ .

$2 \log (x) + 3 \log (y) + 5 \log (z) - 4 \log (a) - (6 \log (b)) - (7 \log (c))$

Step 12.2.2

Multiply $6$ by $- 1$ .

$2 \log (x) + 3 \log (y) + 5 \log (z) - 4 \log (a) - 6 \log (b) - (7 \log (c))$

Step 12.2.3

Multiply $7$ by $- 1$ .

$2 \log (x) + 3 \log (y) + 5 \log (z) - 4 \log (a) - 6 \log (b) - 7 \log (c)$