Simplify/Condense log base 3 of 9x^4- log base 3 of (3x)^2

$\log_{3} (9 x^{4}) - \log_{3} ({(3 x)}^{2})$

Step 1

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

$\log_{3} (\frac{9 x^{4}}{{(3 x)}^{2}})$

Step 2

Simplify the denominator.

Apply the product rule to $3 x$ .

$\log_{3} (\frac{9 x^{4}}{3^{2} x^{2}})$

Raise $3$ to the power of $2$ .

$\log_{3} (\frac{9 x^{4}}{9 x^{2}})$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $9$ .

Cancel the common factor.

$\log_{3} (\frac{9 x^{4}}{9 x^{2}})$

Rewrite the expression.

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

Cancel the common factor of $x^{4}$ and $x^{2}$ .

Factor $x^{2}$ out of $x^{4}$ .

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

Cancel the common factors.

Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\log_{3} (\frac{x^{2}}{1})$

Divide $x^{2}$ by $1$ .

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