Precalculus Examples

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

Step 1

Simplify $3 \log (x^{4} y^{3})$ by moving $3$ inside the logarithm.

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

Step 2

Rewrite using the commutative property of multiplication.

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

Step 3

Apply the product rule to $x^{4} y^{3}$ .

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

Step 4

Multiply the exponents in ${(x^{4})}^{3}$ .

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Step 4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2

Multiply $4$ by $3$ .

$3 \log (x^{12} {(y^{3})}^{3}) \log (\frac{1}{x}) - 4 \log (y)$

Step 5

Multiply the exponents in ${(y^{3})}^{3}$ .

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Step 5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 \log (x^{12} y^{3 \cdot 3}) \log (\frac{1}{x}) - 4 \log (y)$

Step 5.2

Multiply $3$ by $3$ .

$3 \log (x^{12} y^{9}) \log (\frac{1}{x}) - 4 \log (y)$

Step 6

Simplify $3 \log (x^{12} y^{9})$ by moving $3$ inside the logarithm.

$\log ({(x^{12} y^{9})}^{3}) \log (\frac{1}{x}) - 4 \log (y)$

Step 7

Apply the product rule to $x^{12} y^{9}$ .

$\log ({(x^{12})}^{3} {(y^{9})}^{3}) \log (\frac{1}{x}) - 4 \log (y)$

Step 8

Multiply the exponents in ${(x^{12})}^{3}$ .

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Step 8.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (x^{12 \cdot 3} {(y^{9})}^{3}) \log (\frac{1}{x}) - 4 \log (y)$

Step 8.2

Multiply $12$ by $3$ .

$\log (x^{36} {(y^{9})}^{3}) \log (\frac{1}{x}) - 4 \log (y)$

Step 9

Multiply the exponents in ${(y^{9})}^{3}$ .

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Step 9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (x^{36} y^{9 \cdot 3}) \log (\frac{1}{x}) - 4 \log (y)$

Step 9.2

Multiply $9$ by $3$ .

$\log (x^{36} y^{27}) \log (\frac{1}{x}) - 4 \log (y)$

Step 10

Simplify $- 4 \log (y)$ by moving $4$ inside the logarithm.

$\log (x^{36} y^{27}) \log (\frac{1}{x}) - \log (y^{4})$