Precalculus Examples

$\log_{3} (2) - 2 \log_{3} (5 x) + 3 \log_{3} (x)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $- 2 \log_{3} (5 x)$ by moving $2$ inside the logarithm.

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

Step 1.2

Apply the product rule to $5 x$ .

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

Step 1.3

Raise $5$ to the power of $2$ .

$\log_{3} (2) - \log_{3} (25 x^{2}) + 3 \log_{3} (x)$

Step 1.4

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

$\log_{3} (2) - \log_{3} (25 x^{2}) + \log_{3} (x^{3})$

Step 2

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

$\log_{3} (\frac{2}{25 x^{2}}) + \log_{3} (x^{3})$

Step 3

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

$\log_{3} (\frac{2}{25 x^{2}} x^{3})$

Step 4

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

Tap for more steps...

Step 4.1

Factor $x^{2}$ out of $25 x^{2}$ .

$\log_{3} (\frac{2}{x^{2} \cdot 25} x^{3})$

Step 4.2

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

$\log_{3} (\frac{2}{x^{2} \cdot 25} (x^{2} x))$

Step 4.3

Cancel the common factor.

$\log_{3} (\frac{2}{x^{2} \cdot 25} (x^{2} x))$

Step 4.4

Rewrite the expression.

$\log_{3} (\frac{2}{25} x)$

Step 5

Combine $\frac{2}{25}$ and $x$ .

$\log_{3} (\frac{2 x}{25})$