Precalculus Examples

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

Step 1

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

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

Step 2

Simplify $2 \log_{4} (x + 1)$ by moving $2$ inside the logarithm.

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

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 4

Combine $\frac{x}{y}$ and ${(x + 1)}^{2}$ .

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

Step 5

Simplify $\frac{1}{3} \log_{4} (\frac{x {(x + 1)}^{2}}{y})$ by moving $\frac{1}{3}$ inside the logarithm.

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

Step 6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.1

Apply the product rule to $\frac{x {(x + 1)}^{2}}{y}$ .

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

Step 6.2

Apply the product rule to $x {(x + 1)}^{2}$ .

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

Step 7

Multiply the exponents in ${({(x + 1)}^{2})}^{\frac{1}{3}}$ .

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Step 7.1

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

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

Step 7.2

Combine $2$ and $\frac{1}{3}$ .

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