Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Multiply $- \frac{1}{2} \log (y)$ .

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Step 1.1.1

Reorder $\log (y)$ and $\frac{1}{2}$ .

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

Step 1.1.2

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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