Evaluate 2 log base 5 of 3- log base 5 of 45

$2 \log_{5} (3) - \log_{5} (45)$

Simplify each term.

Simplify $2 \log_{5} (3)$ by moving $2$ inside the logarithm.

$\log_{5} (3^{2}) - \log_{5} (45)$

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

$\log_{5} (9) - \log_{5} (45)$

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

$\log_{5} (\frac{9}{45})$

Reduce the expression $\frac{9}{45}$ by cancelling the common factors.

Factor $9$ out of $9$ .

$\log_{5} (\frac{9 (1)}{45})$

Factor $9$ out of $45$ .

$\log_{5} (\frac{9 \cdot 1}{9 \cdot 5})$

Cancel the common factor.

$\log_{5} (\frac{9 \cdot 1}{9 \cdot 5})$

Rewrite the expression.

$\log_{5} (\frac{1}{5})$

Rewrite $\frac{1}{5}$ as $1 \cdot 5^{- 1}$ .

$\log_{5} (1 \cdot 5^{- 1})$

Rewrite $\log_{5} (1 \cdot 5^{- 1})$ as $\log_{5} (1) + \log_{5} (5^{- 1})$ .

$\log_{5} (1) + \log_{5} (5^{- 1})$

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

$\log_{5} (1) - \log_{5} (5)$

Logarithm base $5$ of $5$ is $1$ .

$\log_{5} (1) - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$\log_{5} (1) - 1$

Logarithm base $5$ of $1$ is $0$ .

$0 - 1$

Subtract $1$ from $0$ .

$- 1$