Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

Cancel the common factor of $x^{\frac{3}{2}}$ .

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Step 4.1

Factor $x^{\frac{3}{2}}$ out of $x^{2}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Multiply $x^{\frac{3}{4}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.1

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

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

Step 5.3.2

Multiply $2$ by $2$ .

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 5.5

Add $3$ and $2$ .

$\log_{2} (x^{\frac{5}{4}})$