Precalculus Examples

$\ln (6 e) - \ln (42) + \ln (7 e)$

Step 1

Rewrite $\ln (6 e)$ as $\ln (6) + \ln (e)$ .

$\ln (6) + \ln (e) - \ln (42) + \ln (7 e)$

Step 2

The natural logarithm of $e$ is $1$ .

$\ln (6) + 1 - \ln (42) + \ln (7 e)$

Step 3

Rewrite $\ln (7 e)$ as $\ln (7) + \ln (e)$ .

$\ln (6) + 1 - \ln (42) + \ln (7) + \ln (e)$

Step 4

The natural logarithm of $e$ is $1$ .

$\ln (6) + 1 - \ln (42) + \ln (7) + 1$

Step 5

Simplify each term.

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Step 5.1

Rewrite $\ln (6)$ as $\ln (2 (3))$ .

$\ln (2 (3)) + 1 - \ln (42) + \ln (7) + 1$

Step 5.2

Rewrite $\ln (2 (3))$ as $\ln (2) + \ln (3)$ .

$\ln (2) + \ln (3) + 1 - \ln (42) + \ln (7) + 1$

Step 5.3

Rewrite $\ln (42)$ as $\ln (2 \cdot 3 \cdot 7)$ .

$\ln (2) + \ln (3) + 1 - \ln (2 \cdot 3 \cdot 7) + \ln (7) + 1$

Step 5.4

Rewrite $\ln (2 \cdot 3 \cdot 7)$ as $\ln (2 \cdot 3) + \ln (7)$ .

$\ln (2) + \ln (3) + 1 - (\ln (2 \cdot 3) + \ln (7)) + \ln (7) + 1$

Step 5.5

Rewrite $\ln (2 \cdot 3)$ as $\ln (2) + \ln (3)$ .

$\ln (2) + \ln (3) + 1 - (\ln (2) + \ln (3) + \ln (7)) + \ln (7) + 1$

Step 5.6

Apply the distributive property.

$\ln (2) + \ln (3) + 1 - \ln (2) - \ln (3) - \ln (7) + \ln (7) + 1$

Step 6

Subtract $\ln (2)$ from $\ln (2)$ .

$0 + \ln (3) + 1 - \ln (3) - \ln (7) + \ln (7) + 1$

Step 7

Subtract $\ln (3)$ from $\ln (3)$ .

$0 + 0 + 1 - \ln (7) + \ln (7) + 1$

Step 8

Add $- \ln (7)$ and $\ln (7)$ .

$0 + 0 + 1 + 0 + 1$

Step 9

Add $0$ and $0$ .

$0 + 1 + 0 + 1$

Step 10

Add $0$ and $1$ .

$1 + 0 + 1$

Step 11

Add $1$ and $0$ .

$1 + 1$

Step 12

Add $1$ and $1$ .

$2$