Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Apply the product rule to $3 x$ .

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

Step 1.3

Multiply $- \frac{1}{3} \log_{2} (4 x)$ .

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Step 1.3.1

Reorder $\log_{2} (4 x)$ and $\frac{1}{3}$ .

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

Step 1.3.2

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

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

Step 1.4

Apply the product rule to $4 x$ .

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

Step 2

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

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

Step 3

Move $x^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 4

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

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Step 4.1

Move $x^{- \frac{1}{3}}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Multiply $3$ by $2$ .

$\log_{2} (\frac{3^{\frac{1}{2}} x^{- \frac{2}{6} + \frac{1}{2} \cdot \frac{3}{3}}}{4^{\frac{1}{3}}})$

Step 4.5.3

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

$\log_{2} (\frac{3^{\frac{1}{2}} x^{- \frac{2}{6} + \frac{3}{2 \cdot 3}}}{4^{\frac{1}{3}}})$

Step 4.5.4

Multiply $2$ by $3$ .

$\log_{2} (\frac{3^{\frac{1}{2}} x^{- \frac{2}{6} + \frac{3}{6}}}{4^{\frac{1}{3}}})$

Step 4.6

Combine the numerators over the common denominator.

$\log_{2} (\frac{3^{\frac{1}{2}} x^{\frac{- 2 + 3}{6}}}{4^{\frac{1}{3}}})$

Step 4.7

Add $- 2$ and $3$ .

$\log_{2} (\frac{3^{\frac{1}{2}} x^{\frac{1}{6}}}{4^{\frac{1}{3}}})$