Precalculus Examples

$\log_{5} (525) - \log_{5} (21)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite $\log_{5} (525)$ as $\log_{5} (3 \cdot 5^{2} \cdot 7)$ .

$\log_{5} (3 \cdot 5^{2} \cdot 7) - \log_{5} (21)$

Step 1.2

Rewrite $\log_{5} (3 \cdot 5^{2} \cdot 7)$ as $\log_{5} (3 \cdot 5^{2}) + \log_{5} (7)$ .

$\log_{5} (3 \cdot 5^{2}) + \log_{5} (7) - \log_{5} (21)$

Step 1.3

Simplify each term.

Tap for more steps...

Step 1.3.1

Rewrite $\log_{5} (3 \cdot 5^{2})$ as $\log_{5} (3) + \log_{5} (5^{2})$ .

$\log_{5} (3) + \log_{5} (5^{2}) + \log_{5} (7) - \log_{5} (21)$

Step 1.3.2

Use logarithm rules to move $2$ out of the exponent.

$\log_{5} (3) + 2 \log_{5} (5) + \log_{5} (7) - \log_{5} (21)$

Step 1.3.3

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

$\log_{5} (3) + 2 \cdot 1 + \log_{5} (7) - \log_{5} (21)$

Step 1.3.4

Multiply $2$ by $1$ .

$\log_{5} (3) + 2 + \log_{5} (7) - \log_{5} (21)$

Step 1.4

Rewrite $\log_{5} (21)$ as $\log_{5} (3 (7))$ .

$\log_{5} (3) + 2 + \log_{5} (7) - \log_{5} (3 (7))$

Step 1.5

Rewrite $\log_{5} (3 (7))$ as $\log_{5} (3) + \log_{5} (7)$ .

$\log_{5} (3) + 2 + \log_{5} (7) - (\log_{5} (3) + \log_{5} (7))$

Step 1.6

Apply the distributive property.

$\log_{5} (3) + 2 + \log_{5} (7) - \log_{5} (3) - \log_{5} (7)$

Step 2

Subtract $\log_{5} (3)$ from $\log_{5} (3)$ .

$0 + 2 + \log_{5} (7) - \log_{5} (7)$

Step 3

Subtract $\log_{5} (7)$ from $\log_{5} (7)$ .

$0 + 2 + 0$

Step 4

Add $0$ and $2$ .

$2 + 0$

Step 5

Add $2$ and $0$ .

$2$