Calculus Examples

$\frac{12}{\ln (12)} - \frac{12^{- 1}}{\ln (12)}$

Step 1

Combine the numerators over the common denominator.

$\frac{12 - 12^{- 1}}{\ln (12)}$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12 - \frac{1}{12}}{\ln (12)}$

Step 3

To write $12$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{12 \cdot \frac{12}{12} - \frac{1}{12}}{\ln (12)}$

Step 4

Combine $12$ and $\frac{12}{12}$ .

$\frac{\frac{12 \cdot 12}{12} - \frac{1}{12}}{\ln (12)}$

Step 5

Combine the numerators over the common denominator.

$\frac{\frac{12 \cdot 12 - 1}{12}}{\ln (12)}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $12$ by $12$ .

$\frac{\frac{144 - 1}{12}}{\ln (12)}$

Step 6.2

Subtract $1$ from $144$ .

$\frac{\frac{143}{12}}{\ln (12)}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\frac{143}{12} \cdot \frac{1}{\ln (12)}$

Step 8

Simplify the denominator.

Tap for more steps...

Step 8.1

Rewrite $\ln (12)$ as $\ln (2^{2} \cdot 3)$ .

$\frac{143}{12} \cdot \frac{1}{\ln (2^{2} \cdot 3)}$

Step 8.2

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

$\frac{143}{12} \cdot \frac{1}{\ln (2^{2}) + \ln (3)}$

Step 8.3

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

$\frac{143}{12} \cdot \frac{1}{2 \ln (2) + \ln (3)}$

Step 9

Multiply $\frac{143}{12}$ by $\frac{1}{2 \ln (2) + \ln (3)}$ .

$\frac{143}{12 (2 \ln (2) + \ln (3))}$