Precalculus Examples

$\log_{3} (9) - \log_{5} (5^{3}) + \log_{7} (7^{2}) - \log_{11} (1)$

Step 1

Simplify each term.

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Step 1.1

Logarithm base $3$ of $9$ is $2$ .

$2 - \log_{5} (5^{3}) + \log_{7} (7^{2}) - \log_{11} (1)$

Step 1.2

Use logarithm rules to move $3$ out of the exponent.

$2 - (3 \log_{5} (5)) + \log_{7} (7^{2}) - \log_{11} (1)$

Step 1.3

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

$2 - (3 \cdot 1) + \log_{7} (7^{2}) - \log_{11} (1)$

Step 1.4

Multiply $3$ by $1$ .

$2 - 1 \cdot 3 + \log_{7} (7^{2}) - \log_{11} (1)$

Step 1.5

Multiply $- 1$ by $3$ .

$2 - 3 + \log_{7} (7^{2}) - \log_{11} (1)$

Step 1.6

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

$2 - 3 + 2 \log_{7} (7) - \log_{11} (1)$

Step 1.7

Logarithm base $7$ of $7$ is $1$ .

$2 - 3 + 2 \cdot 1 - \log_{11} (1)$

Step 1.8

Multiply $2$ by $1$ .

$2 - 3 + 2 - \log_{11} (1)$

Step 1.9

Logarithm base $11$ of $1$ is $0$ .

$2 - 3 + 2 - 0$

Step 1.10

Multiply $- 1$ by $0$ .

$2 - 3 + 2 + 0$

Step 2

Simplify by adding and subtracting.

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Step 2.1

Subtract $3$ from $2$ .

$- 1 + 2 + 0$

Step 2.2

Add $- 1$ and $2$ .

$1 + 0$

Step 2.3

Add $1$ and $0$ .

$1$