Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log_{3} (x)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 1.2

Simplify $- 2 \log_{3} (y)$ by moving $2$ inside the logarithm.

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

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

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