Precalculus Examples

$2 \log_{6} (\frac{x}{3}) - 4 \log_{6} (\frac{1}{y})$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{6} (\frac{x}{3})$ by moving $2$ inside the logarithm.

$\log_{6} ({(\frac{x}{3})}^{2}) - 4 \log_{6} (\frac{1}{y})$

Step 1.2

Apply the product rule to $\frac{x}{3}$ .

$\log_{6} (\frac{x^{2}}{3^{2}}) - 4 \log_{6} (\frac{1}{y})$

Step 1.3

Raise $3$ to the power of $2$ .

$\log_{6} (\frac{x^{2}}{9}) - 4 \log_{6} (\frac{1}{y})$

Step 1.4

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

$\log_{6} (\frac{x^{2}}{9}) - \log_{6} ({(\frac{1}{y})}^{4})$

Step 1.5

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

$\log_{6} (\frac{x^{2}}{9}) - \log_{6} (\frac{1^{4}}{y^{4}})$

Step 1.6

One to any power is one.

$\log_{6} (\frac{x^{2}}{9}) - \log_{6} (\frac{1}{y^{4}})$

Step 2

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

$\log_{6} (\frac{\frac{x^{2}}{9}}{\frac{1}{y^{4}}})$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\log_{6} (\frac{x^{2}}{9} y^{4})$

Step 4

Combine $\frac{x^{2}}{9}$ and $y^{4}$ .

$\log_{6} (\frac{x^{2} y^{4}}{9})$