Precalculus Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

Move $3$ to the left of $y$ .

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

Step 2.2

Multiply $- \frac{1}{3} \log (x)$ .

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Step 2.2.1

Reorder $\log (x)$ and $\frac{1}{3}$ .

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

Step 2.2.2

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

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

Step 2.3

Simplify $2 \log (z)$ by moving $2$ inside the logarithm.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine $\frac{3 y}{x^{\frac{1}{3}}}$ and $z^{2}$ .

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