Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Simplify $(\frac{2}{3}) \log_{2} (x)$ by moving $\frac{2}{3}$ inside the logarithm.

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 4.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 4.3

Add $1$ and $2$ .

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

Step 4.4

Divide $3$ by $3$ .

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

Step 5

Simplify $x^{1}$ .

$\log_{2} (\frac{x}{y})$