This Quantum World/Appendix/Exponential function

The exponential function We define the function by requiring that  and   The value of this function is everywhere equal to its slope. Differentiating the first defining equation repeatedly we find that The second defining equation now tells us that for all  The result is a particularly simple Taylor series: Let us check that a well-behaved function satisfies the equation if and only if We will do this by expanding the 's in powers of  and  and compare coefficents. We have and using the binomial expansion we also have that Voilà. The function obviously satisfies and hence So does the function Moreover, implies We gather from this that the functions satisfying form a one-parameter family, the parameter being the real number and that the one-parameter family of functions satisfies , the parameter being the real number  But also defines a one-parameter family of functions that satisfies , the parameter being the positive number  Conclusion: for every real number there is a positive number (and vice versa) such that One of the most important numbers is defined as the number for which that is: : The natural logarithm is defined as the inverse of so Show that Hint: differentiate