Precalculus Examples

$5 \log_{4} (x) + 6 \log_{4} (y) - \frac{1}{3} \cdot \log_{4} (7 z)$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \log_{4} (x)$ by moving $5$ inside the logarithm.

$\log_{4} (x^{5}) + 6 \log_{4} (y) - \frac{1}{3} \cdot \log_{4} (7 z)$

Step 1.2

Simplify $6 \log_{4} (y)$ by moving $6$ inside the logarithm.

$\log_{4} (x^{5}) + \log_{4} (y^{6}) - \frac{1}{3} \cdot \log_{4} (7 z)$

Step 1.3

Multiply $- \frac{1}{3} \log_{4} (7 z)$ .

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Step 1.3.1

Reorder $\log_{4} (7 z)$ and $\frac{1}{3}$ .

$\log_{4} (x^{5}) + \log_{4} (y^{6}) - (\frac{1}{3} \log_{4} (7 z))$

Step 1.3.2

Simplify $\frac{1}{3} \log_{4} (7 z)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{4} (x^{5}) + \log_{4} (y^{6}) - \log_{4} ({(7 z)}^{\frac{1}{3}})$

Step 1.4

Apply the product rule to $7 z$ .

$\log_{4} (x^{5}) + \log_{4} (y^{6}) - \log_{4} (7^{\frac{1}{3}} z^{\frac{1}{3}})$

Step 2

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

$\log_{4} (x^{5} y^{6}) - \log_{4} (7^{\frac{1}{3}} z^{\frac{1}{3}})$

Step 3

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

$\log_{4} (\frac{x^{5} y^{6}}{7^{\frac{1}{3}} z^{\frac{1}{3}}})$