Precalculus Examples

$4 \log_{3} (x) - 8 \log_{3} (y) - \log_{3} (2)$

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \log_{3} (x)$ by moving $4$ inside the logarithm.

$\log_{3} (x^{4}) - 8 \log_{3} (y) - \log_{3} (2)$

Step 1.2

Simplify $- 8 \log_{3} (y)$ by moving $8$ inside the logarithm.

$\log_{3} (x^{4}) - \log_{3} (y^{8}) - \log_{3} (2)$

Step 2

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

$\log_{3} (\frac{x^{4}}{y^{8}}) - \log_{3} (2)$

Step 3

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

$\log_{3} (\frac{\frac{x^{4}}{y^{8}}}{2})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{3} (\frac{x^{4}}{y^{8}} \cdot \frac{1}{2})$

Step 5

Combine.

$\log_{3} (\frac{x^{4} \cdot 1}{y^{8} \cdot 2})$

Step 6

Simplify the expression.

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Step 6.1

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

$\log_{3} (\frac{x^{4}}{y^{8} \cdot 2})$

Step 6.2

Move $2$ to the left of $y^{8}$ .

$\log_{3} (\frac{x^{4}}{2 y^{8}})$