Precalculus Examples

$\ln (3) + \frac{1}{2} \cdot \ln (x) + \ln (y) - \frac{1}{3} \cdot \ln (z)$

Step 1

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

$\frac{1}{2} \cdot \ln (x) + \ln (3 y) - \frac{1}{3} \cdot \ln (z)$

Step 2

Simplify each term.

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Step 2.1

Simplify $\frac{1}{2} \ln (x)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (x^{\frac{1}{2}}) + \ln (3 y) - \frac{1}{3} \cdot \ln (z)$

Step 2.2

Multiply $- \frac{1}{3} \ln (z)$ .

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Step 2.2.1

Reorder $\ln (z)$ and $\frac{1}{3}$ .

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

Step 2.2.2

Simplify $\frac{1}{3} \ln (z)$ by moving $\frac{1}{3}$ inside the logarithm.

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

Step 3

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

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

Step 4

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

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

Step 5

Move $3$ to the left of $x^{\frac{1}{2}}$ .

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

Step 6

Reorder factors in $\ln (\frac{3 x^{\frac{1}{2}} y}{z^{\frac{1}{3}}})$ .

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