# The Natural Exponential Function

**Outline
**1 The definition of the exponential function

2 exp is an exponential function

3 The derivative of e

^{x}

**Definition
**ln, the natural logarithm, is an increasing function with domain (0,∞)
and

range R. Let exp denote the inverse of ln; thus, exp has domain R and

range (0,∞).

**Elementary properties of exp
• **The graph of exp can be obtained directly from the graph of ln.

• We have the important inverse relationships:

ln(exp(x)) = x and exp(ln(x)) = x:

**Euler's number, e
**Recall the definition of e : exp is the number such that ln(e) = 1:

(e≈2.71828)

**Theorem
**exp(r ) = e

^{r}for all rational numbers r :

**Definition
**• Because exp(r ) = e

^{r}for all rational numbers, we will define e

^{x}by

exp(x) for all real numbers x:

• In particular,

ln e

^{x}= x and e

^{ln(x)}= x.

**Theorem (The laws of exponents)
• **e

^{x+y}= e

^{x}e

^{y}

**•**e

^{x-y}= e

^{x}/e

^{y}(for homework)

**•**(ex)

^{r}= e

^{rx}(for homework)

**Problem
**Solve the following equations:

**•**e

^{2x-3}= 8

**•**e

^{2x}+ 2e

^{x}- 8 = 0

**Theorem
**

**Problem
**Find y' in each case:

**Problem
**Solve the following integrals:

**Problem
**Sketch the graph of y = xe

^{-x}.