Precalculus Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $2$ by $2$ .

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

Step 2.4

Cancel the common factor of $x$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

$\ln (y^{2} z^{4}) + \ln (\frac{y}{z})$

Step 3

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

$\ln (y^{2} z^{4} \frac{y}{z})$

Step 4

Cancel the common factor of $z$ .

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Step 4.1

Factor $z$ out of $y^{2} z^{4}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Raise $y$ to the power of $1$ .

$\ln (y^{1} y^{2} z^{3})$

Step 6

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

$\ln (y^{1 + 2} z^{3})$

Step 7

Add $1$ and $2$ .

$\ln (y^{3} z^{3})$