Precalculus Examples

$\log_{3} (18) - \log_{3} (100) - \log_{3} (50)$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\log_{3} (18)$ as $\log_{3} (2 \cdot 3^{2})$ .

$\log_{3} (2 \cdot 3^{2}) - \log_{3} (100) - \log_{3} (50)$

Step 1.2

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

$\log_{3} (2) + \log_{3} (3^{2}) - \log_{3} (100) - \log_{3} (50)$

Step 1.3

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

$\log_{3} (2) + 2 \log_{3} (3) - \log_{3} (100) - \log_{3} (50)$

Step 1.4

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

$\log_{3} (2) + 2 \cdot 1 - \log_{3} (100) - \log_{3} (50)$

Step 1.5

Multiply $2$ by $1$ .

$\log_{3} (2) + 2 - \log_{3} (100) - \log_{3} (50)$

Step 1.6

Rewrite $\log_{3} (100)$ as $\log_{3} (2^{2} \cdot 5^{2})$ .

$\log_{3} (2) + 2 - \log_{3} (2^{2} \cdot 5^{2}) - \log_{3} (50)$

Step 1.7

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

$\log_{3} (2) + 2 - (\log_{3} (2^{2}) + \log_{3} (5^{2})) - \log_{3} (50)$

Step 1.8

Simplify each term.

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Step 1.8.1

Expand $\log_{3} (2^{2})$ by moving $2$ outside the logarithm.

$\log_{3} (2) + 2 - (2 \log_{3} (2) + \log_{3} (5^{2})) - \log_{3} (50)$

Step 1.8.2

Expand $\log_{3} (5^{2})$ by moving $2$ outside the logarithm.

$\log_{3} (2) + 2 - (2 \log_{3} (2) + 2 \log_{3} (5)) - \log_{3} (50)$

Step 1.9

Apply the distributive property.

$\log_{3} (2) + 2 - (2 \log_{3} (2)) - (2 \log_{3} (5)) - \log_{3} (50)$

Step 1.10

Multiply $2$ by $- 1$ .

$\log_{3} (2) + 2 - 2 \log_{3} (2) - (2 \log_{3} (5)) - \log_{3} (50)$

Step 1.11

Multiply $2$ by $- 1$ .

$\log_{3} (2) + 2 - 2 \log_{3} (2) - 2 \log_{3} (5) - \log_{3} (50)$

Step 1.12

Rewrite $\log_{3} (50)$ as $\log_{3} (2 \cdot 5^{2})$ .

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

Step 1.13

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

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

Step 1.14

Expand $\log_{3} (5^{2})$ by moving $2$ outside the logarithm.

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

Step 1.15

Apply the distributive property.

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

Step 1.16

Multiply $2$ by $- 1$ .

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

Step 2

Subtract $\log_{3} (2)$ from $\log_{3} (2)$ .

$0 + 2 - 2 \log_{3} (2) - 2 \log_{3} (5) - 2 \log_{3} (5)$

Step 3

Add $0$ and $2$ .

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

Step 4

Subtract $2 \log_{3} (5)$ from $- 2 \log_{3} (5)$ .

$2 - 2 \log_{3} (2) - 4 \log_{3} (5)$