Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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Step 3.1

Factor $9$ out of $9$ .

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

Step 3.2

Factor $9$ out of $45$ .

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

Step 3.3

Cancel the common factor.

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

Step 3.4

Rewrite the expression.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Multiply $- 1$ by $1$ .

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

Step 9

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

$0 - 1$

Step 10

Subtract $1$ from $0$ .

$- 1$