Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Apply the product rule to $\frac{1}{z}$ .

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

Step 1.4

One to any power is one.

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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