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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.
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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
