Simplify/Condense 21 log base 3 of cube root of x+ log base 3 of 9x^2- log base 3 of 9

$21 \log_{3} (\sqrt[3]{x}) + \log_{3} (9 x^{2}) - \log_{3} (9)$

Step 1

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

$21 \log_{3} (\sqrt[3]{x}) + \log_{3} (\frac{9 x^{2}}{9})$

Step 2

Simplify each term.

Simplify $21 \log_{3} (\sqrt[3]{x})$ by moving $21$ inside the logarithm.

$\log_{3} ({\sqrt[3]{x}}^{21}) + \log_{3} (\frac{9 x^{2}}{9})$

Rewrite ${\sqrt[3]{x}}^{21}$ as $x^{7}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\log_{3} ({(x^{\frac{1}{3}})}^{21}) + \log_{3} (\frac{9 x^{2}}{9})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{3} (x^{\frac{1}{3} \cdot 21}) + \log_{3} (\frac{9 x^{2}}{9})$

Combine $\frac{1}{3}$ and $21$ .

$\log_{3} (x^{\frac{21}{3}}) + \log_{3} (\frac{9 x^{2}}{9})$

Cancel the common factor of $21$ and $3$ .

Factor $3$ out of $21$ .

$\log_{3} (x^{\frac{3 \cdot 7}{3}}) + \log_{3} (\frac{9 x^{2}}{9})$

Cancel the common factors.

Factor $3$ out of $3$ .

$\log_{3} (x^{\frac{3 \cdot 7}{3 (1)}}) + \log_{3} (\frac{9 x^{2}}{9})$

Cancel the common factor.

$\log_{3} (x^{\frac{3 \cdot 7}{3 \cdot 1}}) + \log_{3} (\frac{9 x^{2}}{9})$

Rewrite the expression.

$\log_{3} (x^{\frac{7}{1}}) + \log_{3} (\frac{9 x^{2}}{9})$

Divide $7$ by $1$ .

$\log_{3} (x^{7}) + \log_{3} (\frac{9 x^{2}}{9})$

Cancel the common factor of $9$ .

Cancel the common factor.

$\log_{3} (x^{7}) + \log_{3} (\frac{9 x^{2}}{9})$

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

$\log_{3} (x^{7}) + \log_{3} (x^{2})$

Step 3

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

$\log_{3} (x^{7} x^{2})$

Step 4

Multiply $x^{7}$ by $x^{2}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{3} (x^{7 + 2})$

Add $7$ and $2$ .

$\log_{3} (x^{9})$