Simplify/Condense 3 log of x+ log of 4- log of x- log of 6

$3 \log (x) + \log (4) - \log (x) - \log (6)$

Step 1

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

$3 \log (x) + \log (\frac{4}{x}) - \log (6)$

Step 2

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

$3 \log (x) + \log (\frac{\frac{4}{x}}{6})$

Step 3

Simplify each term.

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

$\log (x^{3}) + \log (\frac{\frac{4}{x}}{6})$

Multiply the numerator by the reciprocal of the denominator.

$\log (x^{3}) + \log (\frac{4}{x} \cdot \frac{1}{6})$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\log (x^{3}) + \log (\frac{2 (2)}{x} \cdot \frac{1}{6})$

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Move $3$ to the left of $x$ .

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

Step 4

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

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

Step 5

Cancel the common factor of $x$ .

Factor $x$ out of $x^{3}$ .

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

Factor $x$ out of $3 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

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

Step 7

Move $2$ to the left of $x^{2}$ .

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