Because with that particular case #a=e# the exponent function #a^x# is its own derivative, and because of uniqueness this means it is the only function that remains unchanged when differentiated and integrated.
# d/dx e^x = e^x # , and, #int e^x \ dx = e^x " "(+c) #
#e# is an irrational number (like #pi# or #sqrt(2)#), and like #pi# the number crops up in many natural circumstances (e.g. radioactive decay, temperature cooling, compound interest, probability etc.)
It is also related to #pi#, and #i# the imaginary number by Euler's famous identity:
#e^(ipi) + 1 -= 0#
#e# is also the base used for "Natural" Logarithms,denoted by #lnx#, where:
#lnx=a <=> x=e^a#
And the Natural Logarithm has the property of being the area under the curve #y=1/x#, as
# int_1^x \ 1/t \ dt = lnx => int_1^e \ 1/t \ dt = 1#
#e^x# can be expanded as a Power Series (using Taylor's Theorem) to give;
#e^x = 1+x+x^2/(2!)+x^3/(3!)+x^4/(4!) + ... = sum_(n=0)^oo x^n/(n!)#
which is convergent for all real values of #x#, And so #e# itself can be written as the infinite sum:
#e = 1+1+1/(2!)+1/(3!)+1/(4!) + ... = sum_(n=0)^oo 1/(n!)#
and that #e# is the value of the limit:
# lim_(n rarr oo) (1 + 1/n)^n #
Euler gave an approximation for #e# to 18 decimal places,
#e = 2.718281828459045235 #
