Section 7.2

1. Section 7.2 The Natural Exponential Function

2. THE NATURAL EXPONENTIAL FUNCTION Definition: The inverse of the one-to-one natural logarithmic function is the natural exponential function defined by y = exp(x) if, and only if, ln y = x

3. COMMENTS ON THE NATURAL EXPONENTIAL FUNCTION • exp(ln x) = x and ln(exp x) = x • exp(0) = 1 since ln 1 = 0 • exp(1) = e since ln e = 1 • For any rational number r, ln(er) = r ln e = r. Hence, exp(r) = er for any rational number r.

4. DEFINITION Definition: For all real numbers x, ex = exp(x)

5. COMMENTS ON ex 1. ex = y if, and only if, ln y = x 2. eln x = xx > 0 3. ln(ex) = x for all x

6. PROPERTIES OF THE NATURAL EXPONENTIAL FUNCTION f(x) = ex • It is an increasing continuous function. • Its domain is (−∞, ∞). • Its range is (0, ∞). so the x-axis is a horizontal asymptote of its graph.

7. LAWS OF EXPONENTS If x and y are real numbers and r is rational, then

8. DIFFERENTIATION OF ex

9. ANTIDIFFERENTIATION OF ex