Write as a Single Logarithm log base 7 of 2+2 log base 7 of 6- log base 7 of 9

$\log_{7} (2) + 2 \log_{7} (6) - \log_{7} (9)$

Step 1

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

$2 \log_{7} (6) + \log_{7} (\frac{2}{9})$

Step 2

Simplify each term.

Simplify $2 \log_{7} (6)$ by moving $2$ inside the logarithm.

$\log_{7} (6^{2}) + \log_{7} (\frac{2}{9})$

Raise $6$ to the power of $2$ .

$\log_{7} (36) + \log_{7} (\frac{2}{9})$

Step 3

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

$\log_{7} (36 (\frac{2}{9}))$

Step 4

Cancel the common factor of $9$ .

Factor $9$ out of $36$ .

$\log_{7} (9 (4) \frac{2}{9})$

Cancel the common factor.

$\log_{7} (9 \cdot 4 \frac{2}{9})$

Rewrite the expression.

$\log_{7} (4 \cdot 2)$

Step 5

Multiply $4$ by $2$ .

$\log_{7} (8)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\log_{7} (8)$

Decimal Form:

$1.06862156 \dots$