Precalculus Examples

$\frac{1}{3} \cdot (\log_{8} (y) + 2 \log_{8} (y + 4)) - \log_{8} (y - 1)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{8} (y + 4)$ by moving $2$ inside the logarithm.

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

Step 1.2

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

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

Step 1.3

Simplify $\frac{1}{3} \log_{8} (y {(y + 4)}^{2})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{8} ({(y {(y + 4)}^{2})}^{\frac{1}{3}}) - \log_{8} (y - 1)$

Step 1.4

Apply the product rule to $y {(y + 4)}^{2}$ .

$\log_{8} (y^{\frac{1}{3}} {({(y + 4)}^{2})}^{\frac{1}{3}}) - \log_{8} (y - 1)$

Step 1.5

Multiply the exponents in ${({(y + 4)}^{2})}^{\frac{1}{3}}$ .

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Step 1.5.1

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

$\log_{8} (y^{\frac{1}{3}} {(y + 4)}^{2 (\frac{1}{3})}) - \log_{8} (y - 1)$

Step 1.5.2

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

$\log_{8} (y^{\frac{1}{3}} {(y + 4)}^{\frac{2}{3}}) - \log_{8} (y - 1)$

Step 2

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

$\log_{8} (\frac{y^{\frac{1}{3}} {(y + 4)}^{\frac{2}{3}}}{y - 1})$