For natural a, we define ba as ∏(b) a times, but for any real a and positive real b, we define ba = exp(a*ln(b)). More on reddit.com

The first definition of exponents is: a^n is a times itself n times. As you correctly pointed out, this only works when n is a natural number. But we can extend this definition by noticing the exponential property, which is that am+n = am ⋅ a^n. This is called the exponential property because it is true for exponents, and only for exponents; and it seems to encapsulate the very idea of exponentiation. Let me explain. The exponential property is true for all exponents. Suppose m and n are natural numbers. Then am+n is a times itself (m+n) times, which is a^m ⋅ a^n. This is only true for exponents. Suppose you have some function f such that f(m+n) = f(m)f(n) for all natural numbers m and n. Then by taking n=1+1+...+1, we see that f(n)=f(1)f(1)...f(1)=f(1)n, so f is an exponential function. The very idea of exponentiation is repeated multiplication. So an+1 is one more a than a^n, hence our rigorous version of the first definition, repeated multiplication, is of the form a1 = a and an+1=an ⋅ a. This formula is just an extension of that. Now, we can use this exponential formula to define all rational exponents. In particular, we get: Zero: We see that a^n = an+0 = a^n ⋅ a0. Therefore, a0 = 1. Negative integers: We see that 1 = a0 = an+(-n) = a^n ⋅ a-n. Therefore a-n = 1/(a^n). Fractions: consider the fraction m/n. We see that m/n+m/n+...+m/n = m if you add m/n to itself n times. So am/n⋅am/n⋅...⋅am/n = am/n+m/n+...+m/n = a^m. Therefore am/n is the n-th root of a^m. There we've extended the definition of exponentiation to all rational numbers. Now all that's left is the irrational reals (and maybe imaginary numbers, but that's for another day. The next thing we notice is that our new definition of exponents is continuous and always increasing for bases a>1 and always decreasing for 0 More on reddit.com