Precalculus Examples

$3 \log (x) + 4 \log (x - 3) - 2 \log (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (x)$ by moving $3$ inside the logarithm.

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

Step 1.2

Simplify $4 \log (x - 3)$ by moving $4$ inside the logarithm.

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

Step 1.3

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

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

Step 2

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

$\log (x^{3} {(x - 3)}^{4}) - \log (x^{2})$

Step 3

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

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

Step 4

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

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Step 4.1

Factor $x^{2}$ out of $x^{3} {(x - 3)}^{4}$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.2.4

Divide $x {(x - 3)}^{4}$ by $1$ .

$\log (x {(x - 3)}^{4})$