Precalculus Examples

$2 \log (2 x) - \log (4 x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log (2 x)$ by moving $2$ inside the logarithm.

$\log ({(2 x)}^{2}) - \log (4 x)$

Step 1.2

Apply the product rule to $2 x$ .

$\log (2^{2} x^{2}) - \log (4 x)$

Step 1.3

Raise $2$ to the power of $2$ .

$\log (4 x^{2}) - \log (4 x)$

Step 2

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

$\log (\frac{4 x^{2}}{4 x})$

Step 3

Cancel the common factor of $4$ .

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Step 3.1

Cancel the common factor.

$\log (\frac{4 x^{2}}{4 x})$

Step 3.2

Rewrite the expression.

$\log (\frac{x^{2}}{x})$

Step 4

Cancel the common factor of $x^{2}$ and $x$ .

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Step 4.1

Factor $x$ out of $x^{2}$ .

$\log (\frac{x \cdot x}{x})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Raise $x$ to the power of $1$ .

$\log (\frac{x \cdot x}{x^{1}})$

Step 4.2.2

Factor $x$ out of $x^{1}$ .

$\log (\frac{x \cdot x}{x \cdot 1})$

Step 4.2.3

Cancel the common factor.

$\log (\frac{x \cdot x}{x \cdot 1})$

Step 4.2.4

Rewrite the expression.

$\log (\frac{x}{1})$

Step 4.2.5

Divide $x$ by $1$ .

$\log (x)$