Chapter 11.6, Problem 1CFU

To determine

The explanation of why $\ln e=1$

Explanation of Solution

Now let us take,

  $\ln e=x\mathrm{......................................}(1)$

Now we know that if,

  $\begin{array}{l}{\log }_{a}b=c,\\ then,\\ a^c=b.\end{array}$

Also we know that,

  $\ln e={\log }_{e}e$

Thus,

  $\begin{array}{l}\ln e={\log }_{e}e=x\\ therefore,\\ e^x=e\\ or,\\ e^1=e\\ Therefore,\\ x=1.\end{array}$

Now putting the value of $x$ in equation (1), we get,

  $\ln e=1$