Algebra Examples

$\log_{3} (a^{7}) + (\log_{3} (a^{4}) - 2 \cdot \log_{3} (b))$

Step 1

Remove parentheses.

$\log_{3} (a^{7}) + \log_{3} (a^{4}) - 2 \cdot \log_{3} (b)$

Step 2

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

$\log_{3} (a^{7} a^{4}) - 2 \cdot \log_{3} (b)$

Step 3

Simplify each term.

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Step 3.1

Multiply $a^{7}$ by $a^{4}$ by adding the exponents.

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Step 3.1.1

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

$\log_{3} (a^{7 + 4}) - 2 \cdot \log_{3} (b)$

Step 3.1.2

Add $7$ and $4$ .

$\log_{3} (a^{11}) - 2 \cdot \log_{3} (b)$

Step 3.2

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

$\log_{3} (a^{11}) - \log_{3} (b^{2})$

Step 4

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

$\log_{3} (\frac{a^{11}}{b^{2}})$