Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Multiply $- \frac{1}{4} \log_{6} (w)$ .

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Step 1.2.1

Reorder $\log_{6} (w)$ and $\frac{1}{4}$ .

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

Step 1.2.2

Simplify $\frac{1}{4} \log_{6} (w)$ by moving $\frac{1}{4}$ inside the logarithm.

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

Step 1.3

Simplify $7 \log_{6} (y)$ by moving $7$ inside the logarithm.

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

Step 2

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

$\log_{6} (\frac{z^{4}}{w^{\frac{1}{4}}}) + \log_{6} (y^{7})$

Step 3

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

$\log_{6} (\frac{z^{4}}{w^{\frac{1}{4}}} y^{7})$

Step 4

Combine $\frac{z^{4}}{w^{\frac{1}{4}}}$ and $y^{7}$ .

$\log_{6} (\frac{z^{4} y^{7}}{w^{\frac{1}{4}}})$