Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

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

Step 3.4.2

Multiply $3$ by $2$ .

$\log_{2} (\frac{x^{\frac{2}{6} + \frac{1}{2} \cdot \frac{3}{3}}}{y})$

Step 3.4.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$\log_{2} (\frac{x^{\frac{2}{6} + \frac{3}{2 \cdot 3}}}{y})$

Step 3.4.4

Multiply $2$ by $3$ .

$\log_{2} (\frac{x^{\frac{2}{6} + \frac{3}{6}}}{y})$

Step 3.5

Combine the numerators over the common denominator.

$\log_{2} (\frac{x^{\frac{2 + 3}{6}}}{y})$

Step 3.6

Add $2$ and $3$ .

$\log_{2} (\frac{x^{\frac{5}{6}}}{y})$