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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.