Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

$\ln (x^{3 \cdot 3} y^{3}) + 2 \ln (y z^{2})$

Step 1.3.2

Multiply $3$ by $3$ .

$\ln (x^{9} y^{3}) + 2 \ln (y z^{2})$

Step 1.4

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

$\ln (x^{9} y^{3}) + \ln ({(y z^{2})}^{2})$

Step 1.5

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

$\ln (x^{9} y^{3}) + \ln (y^{2} {(z^{2})}^{2})$

Step 1.6

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

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Step 1.6.1

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

$\ln (x^{9} y^{3}) + \ln (y^{2} z^{2 \cdot 2})$

Step 1.6.2

Multiply $2$ by $2$ .

$\ln (x^{9} y^{3}) + \ln (y^{2} z^{4})$

Step 2

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

$\ln (x^{9} y^{3} (y^{2} z^{4}))$

Step 3

Multiply $y^{3}$ by $y^{2}$ by adding the exponents.

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Step 3.1

Move $y^{2}$ .

$\ln (x^{9} (y^{2} y^{3}) z^{4})$

Step 3.2

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

$\ln (x^{9} y^{2 + 3} z^{4})$

Step 3.3

Add $2$ and $3$ .

$\ln (x^{9} y^{5} z^{4})$