Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $\frac{x^{2}}{3} x^{3}$ .

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Step 4.1

Combine $\frac{x^{2}}{3}$ and $x^{3}$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Add $2$ and $3$ .

$\log_{2} (\frac{x^{5}}{3})$