Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \log_{b} (x - 2)$ by moving $4$ inside the logarithm.

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

Step 1.2

Multiply $- \frac{1}{2} \log_{b} (y)$ .

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Step 1.2.1

Reorder $\log_{b} (y)$ and $\frac{1}{2}$ .

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

Step 1.2.2

Simplify $\frac{1}{2} \log_{b} (y)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

Multiply ${(x - 2)}^{4}$ by $1$ .

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