Understanding the Natural Exponential Function and Its Properties
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The Natural Exponential Function, defined as ( y = exp(x) ) with the inverse relationship to the natural logarithm, is a fundamental mathematical concept. Key properties include that ( exp(0) = 1 ) and ( exp(1) = e ). It is an increasing continuous function with a domain of ( (-infty, infty) ) and a range of ( (0, infty) ). The function adheres to the laws of exponents, and its differential and integral forms are crucial for calculus applications. This function is foundational in various scientific fields.
Understanding the Natural Exponential Function and Its Properties
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Presentation Transcript
Section 7.2 The Natural Exponential Function
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
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.
DEFINITION Definition: For all real numbers x, ex = exp(x)
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
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.
LAWS OF EXPONENTS If x and y are real numbers and r is rational, then
