Precalculus Examples

$\frac{1}{5} \cdot (3 \log_{b} (x) - 4 \log_{b} (y))$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{1}{5} \cdot (\log_{b} (x^{3}) - 4 \log_{b} (y))$

Step 1.2

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

$\frac{1}{5} \cdot (\log_{b} (x^{3}) - \log_{b} (y^{4}))$

Step 2

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

$\frac{1}{5} \cdot \log_{b} (\frac{x^{3}}{y^{4}})$

Step 3

Simplify $\frac{1}{5} \log_{b} (\frac{x^{3}}{y^{4}})$ by moving $\frac{1}{5}$ inside the logarithm.

$\log_{b} ({(\frac{x^{3}}{y^{4}})}^{\frac{1}{5}})$

Step 4

Apply the product rule to $\frac{x^{3}}{y^{4}}$ .

$\log_{b} (\frac{{(x^{3})}^{\frac{1}{5}}}{{(y^{4})}^{\frac{1}{5}}})$

Step 5

Multiply the exponents in ${(x^{3})}^{\frac{1}{5}}$ .

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Step 5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{b} (\frac{x^{3 (\frac{1}{5})}}{{(y^{4})}^{\frac{1}{5}}})$

Step 5.2

Combine $3$ and $\frac{1}{5}$ .

$\log_{b} (\frac{x^{\frac{3}{5}}}{{(y^{4})}^{\frac{1}{5}}})$

Step 6

Multiply the exponents in ${(y^{4})}^{\frac{1}{5}}$ .

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Step 6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{b} (\frac{x^{\frac{3}{5}}}{y^{4 (\frac{1}{5})}})$

Step 6.2

Combine $4$ and $\frac{1}{5}$ .

$\log_{b} (\frac{x^{\frac{3}{5}}}{y^{\frac{4}{5}}})$