Precalculus Examples

$\frac{1}{4} \cdot \log_{b} (x) - 2 \log_{b} (5) - 10 \log_{b} (y)$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{b} (x^{\frac{1}{4}}) - 2 \log_{b} (5) - 10 \log_{b} (y)$

Step 1.2

Simplify $- 2 \log_{b} (5)$ by moving $2$ inside the logarithm.

$\log_{b} (x^{\frac{1}{4}}) - \log_{b} (5^{2}) - 10 \log_{b} (y)$

Step 1.3

Raise $5$ to the power of $2$ .

$\log_{b} (x^{\frac{1}{4}}) - \log_{b} (25) - 10 \log_{b} (y)$

Step 1.4

Simplify $- 10 \log_{b} (y)$ by moving $10$ inside the logarithm.

$\log_{b} (x^{\frac{1}{4}}) - \log_{b} (25) - \log_{b} (y^{10})$

Step 2

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

$\log_{b} (\frac{x^{\frac{1}{4}}}{25}) - \log_{b} (y^{10})$

Step 3

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

$\log_{b} (\frac{\frac{x^{\frac{1}{4}}}{25}}{y^{10}})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{b} (\frac{x^{\frac{1}{4}}}{25} \cdot \frac{1}{y^{10}})$

Step 5

Combine.

$\log_{b} (\frac{x^{\frac{1}{4}} \cdot 1}{25 y^{10}})$

Step 6

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

$\log_{b} (\frac{x^{\frac{1}{4}}}{25 y^{10}})$