Devin's Reflection

Definition of the Natural Exponential Function

The inverse function of the natural logarithmic function f(x) = ln x is called the natural exponential function and is denoted by f^-1(x) = e^x, that is, y = e^x if and only if x = lny.

The inverse relationship between the natural logarithmic function and the natural exponential function can be summarized as follows.

Ln(e^x) = x and e^lnx = x

Ex. 9 = e^x-3

Ln3 = ln(e^x-3)

Ln9 = x-3

X = 3 + ln9

Theorem 5.10 Operations with exponential functions

Let a and b be any real numbers.

1. E^a e^b = e^(a+b)

2. E^a / e^b = e^(a-b)

Properties of the Natural Exponential Function

1. The domain of f(x) = e^x is (-oo,00), and the range is (0, oo)/

2. The function f(x) = e^x is continuous, increasing, and one to one on its entire domain.

3. The graph of f(x) = e^x is concave upward on its entire domain.

4. Lim e^x = 0 and lim e^x = oo