Algebra Examples

$2 \log_{3} (9) + \log_{3} (7) - \log_{3} (3)$

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

Logarithm base $3$ of $9$ is $2$ .

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

Step 2.2

Multiply $2$ by $2$ .

$4 + \log_{3} (\frac{7}{3})$

Step 2.3

Rewrite $\frac{7}{3}$ as $7 \cdot 3^{- 1}$ .

$4 + \log_{3} (7 \cdot 3^{- 1})$

Step 2.4

Rewrite $\log_{3} (7 \cdot 3^{- 1})$ as $\log_{3} (7) + \log_{3} (3^{- 1})$ .

$4 + \log_{3} (7) + \log_{3} (3^{- 1})$

Step 2.5

Use logarithm rules to move $- 1$ out of the exponent.

$4 + \log_{3} (7) - \log_{3} (3)$

Step 2.6

Logarithm base $3$ of $3$ is $1$ .

$4 + \log_{3} (7) - 1 \cdot 1$

Step 2.7

Multiply $- 1$ by $1$ .

$4 + \log_{3} (7) - 1$

Step 3

Subtract $1$ from $4$ .

$3 + \log_{3} (7)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$3 + \log_{3} (7)$

Decimal Form:

$4.77124374 \dots$