Precalculus Examples

$4 \ln (2 x^{3}) - 2 \ln (x y) + 3 \ln (x^{2} y^{3})$

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \ln (2 x^{3})$ by moving $4$ inside the logarithm.

$\ln ({(2 x^{3})}^{4}) - 2 \ln (x y) + 3 \ln (x^{2} y^{3})$

Step 1.2

Apply the product rule to $2 x^{3}$ .

$\ln (2^{4} {(x^{3})}^{4}) - 2 \ln (x y) + 3 \ln (x^{2} y^{3})$

Step 1.3

Raise $2$ to the power of $4$ .

$\ln (16 {(x^{3})}^{4}) - 2 \ln (x y) + 3 \ln (x^{2} y^{3})$

Step 1.4

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

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Step 1.4.1

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

$\ln (16 x^{3 \cdot 4}) - 2 \ln (x y) + 3 \ln (x^{2} y^{3})$

Step 1.4.2

Multiply $3$ by $4$ .

$\ln (16 x^{12}) - 2 \ln (x y) + 3 \ln (x^{2} y^{3})$

Step 1.5

Simplify $- 2 \ln (x y)$ by moving $2$ inside the logarithm.

$\ln (16 x^{12}) - \ln ({(x y)}^{2}) + 3 \ln (x^{2} y^{3})$

Step 1.6

Apply the product rule to $x y$ .

$\ln (16 x^{12}) - \ln (x^{2} y^{2}) + 3 \ln (x^{2} y^{3})$

Step 1.7

Simplify $3 \ln (x^{2} y^{3})$ by moving $3$ inside the logarithm.

$\ln (16 x^{12}) - \ln (x^{2} y^{2}) + \ln ({(x^{2} y^{3})}^{3})$

Step 1.8

Apply the product rule to $x^{2} y^{3}$ .

$\ln (16 x^{12}) - \ln (x^{2} y^{2}) + \ln ({(x^{2})}^{3} {(y^{3})}^{3})$

Step 1.9

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 1.9.1

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

$\ln (16 x^{12}) - \ln (x^{2} y^{2}) + \ln (x^{2 \cdot 3} {(y^{3})}^{3})$

Step 1.9.2

Multiply $2$ by $3$ .

$\ln (16 x^{12}) - \ln (x^{2} y^{2}) + \ln (x^{6} {(y^{3})}^{3})$

Step 1.10

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

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Step 1.10.1

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

$\ln (16 x^{12}) - \ln (x^{2} y^{2}) + \ln (x^{6} y^{3 \cdot 3})$

Step 1.10.2

Multiply $3$ by $3$ .

$\ln (16 x^{12}) - \ln (x^{2} y^{2}) + \ln (x^{6} y^{9})$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{16 x^{12}}{x^{2} y^{2}}) + \ln (x^{6} y^{9})$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (\frac{16 x^{12}}{x^{2} y^{2}} (x^{6} y^{9}))$

Step 4

Cancel the common factor of $x^{2} y^{2}$ .

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Step 4.1

Factor $x^{2} y^{2}$ out of $x^{6} y^{9}$ .

$\ln (\frac{16 x^{12}}{x^{2} y^{2}} (x^{2} y^{2} (x^{4} y^{7})))$

Step 4.2

Cancel the common factor.

$\ln (\frac{16 x^{12}}{x^{2} y^{2}} (x^{2} y^{2} (x^{4} y^{7})))$

Step 4.3

Rewrite the expression.

$\ln (16 x^{12} (x^{4} y^{7}))$

Step 5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\ln (16 x^{4 + 12} y^{7})$

Step 6

Add $4$ and $12$ .

$\ln (16 x^{16} y^{7})$