Precalculus Examples

$4 \log (x) - 5 \log (y) - \frac{1}{2} \cdot \log (z)$

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \log (x)$ by moving $4$ inside the logarithm.

$\log (x^{4}) - 5 \log (y) - \frac{1}{2} \cdot \log (z)$

Step 1.2

Simplify $- 5 \log (y)$ by moving $5$ inside the logarithm.

$\log (x^{4}) - \log (y^{5}) - \frac{1}{2} \cdot \log (z)$

Step 1.3

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

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Step 1.3.1

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

$\log (x^{4}) - \log (y^{5}) - (\frac{1}{2} \log (z))$

Step 1.3.2

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

$\log (x^{4}) - \log (y^{5}) - \log (z^{\frac{1}{2}})$

Step 2

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

$\log (\frac{x^{4}}{y^{5}}) - \log (z^{\frac{1}{2}})$

Step 3

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

$\log (\frac{\frac{x^{4}}{y^{5}}}{z^{\frac{1}{2}}})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x^{4}}{y^{5}} \cdot \frac{1}{z^{\frac{1}{2}}})$

Step 5

Combine.

$\log (\frac{x^{4} \cdot 1}{y^{5} z^{\frac{1}{2}}})$

Step 6

Multiply $x^{4}$ by $1$ .

$\log (\frac{x^{4}}{y^{5} z^{\frac{1}{2}}})$