Precalculus Examples

$\log_{3} (10) - \log_{3} (180) + \log_{3} (6)$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\log_{3} (10)$ as $\log_{3} (2 (5))$ .

$\log_{3} (2 (5)) - \log_{3} (180) + \log_{3} (6)$

Step 1.2

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

$\log_{3} (2) + \log_{3} (5) - \log_{3} (180) + \log_{3} (6)$

Step 1.3

Rewrite $\log_{3} (180)$ as $\log_{3} (2^{2} \cdot 3^{2} \cdot 5)$ .

$\log_{3} (2) + \log_{3} (5) - \log_{3} (2^{2} \cdot 3^{2} \cdot 5) + \log_{3} (6)$

Step 1.4

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

$\log_{3} (2) + \log_{3} (5) - (\log_{3} (2^{2} \cdot 3^{2}) + \log_{3} (5)) + \log_{3} (6)$

Step 1.5

Simplify each term.

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Step 1.5.1

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

$\log_{3} (2) + \log_{3} (5) - (\log_{3} (2^{2}) + \log_{3} (3^{2}) + \log_{3} (5)) + \log_{3} (6)$

Step 1.5.2

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

$\log_{3} (2) + \log_{3} (5) - (\log_{3} (2^{2}) + 2 \log_{3} (3) + \log_{3} (5)) + \log_{3} (6)$

Step 1.5.3

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

$\log_{3} (2) + \log_{3} (5) - (\log_{3} (2^{2}) + 2 \cdot 1 + \log_{3} (5)) + \log_{3} (6)$

Step 1.5.4

Multiply $2$ by $1$ .

$\log_{3} (2) + \log_{3} (5) - (\log_{3} (2^{2}) + 2 + \log_{3} (5)) + \log_{3} (6)$

Step 1.5.5

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

$\log_{3} (2) + \log_{3} (5) - (2 \log_{3} (2) + 2 + \log_{3} (5)) + \log_{3} (6)$

Step 1.6

Apply the distributive property.

$\log_{3} (2) + \log_{3} (5) - (2 \log_{3} (2)) - 1 \cdot 2 - \log_{3} (5) + \log_{3} (6)$

Step 1.7

Simplify.

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Step 1.7.1

Multiply $2$ by $- 1$ .

$\log_{3} (2) + \log_{3} (5) - 2 \log_{3} (2) - 1 \cdot 2 - \log_{3} (5) + \log_{3} (6)$

Step 1.7.2

Multiply $- 1$ by $2$ .

$\log_{3} (2) + \log_{3} (5) - 2 \log_{3} (2) - 2 - \log_{3} (5) + \log_{3} (6)$

Step 1.8

Rewrite $\log_{3} (6)$ as $\log_{3} (2 (3))$ .

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

Step 1.9

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

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

Step 1.10

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

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

Step 2

Add $\log_{3} (2)$ and $\log_{3} (2)$ .

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

Step 3

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

$2 \log_{3} (2) + 0 - 2 \log_{3} (2) - 2 + 1$

Step 4

Add $2 \log_{3} (2)$ and $0$ .

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

Step 5

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

$0 - 2 + 1$

Step 6

Subtract $2$ from $0$ .

$- 2 + 1$

Step 7

Add $- 2$ and $1$ .

$- 1$