Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Simplify $- 3 \log (w)$ by moving $3$ inside the logarithm.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Combine.

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

Step 7

Multiply $x^{2}$ by $1$ .

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