MATH 12002 - CALCULUS I 5.4: General Exponential & Logarithmic - - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I §5.4: General Exponential & Logarithmic Functions

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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Exponential Functions

An exponential function is a function of the form f (x) = ax, where a > 0, a = 1 is a constant. For example, f (x) = ex, f (x) = 2x, and f (x) = (1

2)x

are all exponential functions. Contrast these with power functions of the form f (x) = xn, where n is a constant, such as f (x) = x2, f (x) = x1/2, or f (x) = xe.

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Exponential Functions

If a > 0 and r = m/n is a rational number (m, n integers, and n > 0) we know that “ar” means am/n =

n

√ am. The meaning of ax when x is irrational (2

√ 2, for example), is less clear.

We could define ax using rational approximations to x: ax = lim

r→x

r rational

ar. It is more convenient, however, to define ax in terms of ex. We have defined ln x = x

1 1 t dt and ex as the inverse of ln x.

Hence for z > 0, eln z = z. In particular, ax = eln(ax) = ex ln a, and so we have for any constant a > 0, ax = e(ln a)x.

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Properties and Graphs

Let a be a constant, a > 0 and a = 1, and let f (x) = ax. f (x) = ax > 0 for all real numbers x. f (0) = a0 = 1, and so the point (0, 1) is on the graph. f (1) = a1 = a, and so the point (1, a) is on the graph. f (−1) = a−1 = 1

a, and so the point (−1, 1/a) is on the graph.

The points (0, 1), (1, a), and (−1, 1/a) are useful reference points for drawing the graph of f (x) = ax and for comparing graphs for different values of a. Other basic properties depend on whether a > 1 or a < 1.

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Properties and Graphs

Let a be a constant, a > 1 , and let f (x) = ax. f (x) is always increasing. f (x) is always concave up. lim

x→+∞ ax = +∞.

lim

x→−∞ ax = 0, so y = 0 is a horizontal asymptote at −∞.

The graph of f (x) = ax has the following shape: §5.4 Figure 1

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Properties and Graphs

Let a be a constant, 0 < a < 1 , and let f (x) = ax. f (x) is always decreasing. f (x) is always concave up. lim

x→+∞ ax = 0, so y = 0 is a horizontal asymptote at +∞.

lim

x→−∞ ax = +∞.

The graph of f (x) = ax has the following shape: §5.4 Figure 2

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General Logarithms

Let a > 0 and a = 1. Since f (x) = ax is either always increasing or always decreasing, it is a one-to-one function, hence has an inverse. The inverse of f (x) = ax is the logarithm with base a, denoted loga x. Notes: f (x) = ax has domain (−∞, ∞) and range (0, ∞). f −1(x) = loga x has domain (0, ∞) and range (−∞, ∞). loga x = y ⇐ ⇒ ay = x. ln x = loge x. log10 x is often denoted log x. The general logarithm loga x turns out to be a constant multiple of ln x, so for our purposes is not very important.

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General Logarithms

Finally, we relate loga x and the natural logarithm ln x. Set loga x = y. We then have x = ay and taking the natural logarithm of both sides, we get ln x = ln(ay) = y ln a. Hence y = ln x/ ln a; that is loga x = 1 ln a ln x. With the formula we derived earlier, ax = e(ln a)x, we can now convert any general exponential or logarithmic function to a natural exponential or logarithmic function.

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