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7.1 The Logarithm Defined as an Integral What is ex, ln x?
We have seen and used the logarithmic function ln x and exponential function ex^ many times. But, what are they? How did we describe ex? Intuitive and informal
- e is some constant: e = 2. 7182821828 ....
- en^ when n is an integer; e.g. e^2 = e · e etc.
- er^ with r being a rational number, e.g. e^23 = (e^2 )^13.
- When x is irrational, the precise meaning of ex^ is not so clear.
Then, ln x is defined as the inverse of ex. Also, we claimed without proof that
(ex)′^ = ex, and (ln x)′^ =^1 x
In this chapter, we will give a rigorous approach to the definitions and properties of these functions, and we study a wide range of applied problems in which they play a role. Definition of the Natural Logarithm Function
To understand this function, we want to study its
- range
- graph (monotonicity, concavity etc.)
- derivative
- inverse
- other properties Domain of ln x: x > 0
The function is not defined for x ≤ 0.
The Derivative of y = ln x
By the fundamental theorem of calculus, d dx (ln x) = d dx
(∫ (^) x
1
t dt
x when x > 0.
If u (x) is differentiable and positive, then by the Chain Rule,
d dx (ln u (x)) =
u (x)
du dx
u′^ (x) u (x) e.g. d dx (ln bx) =
bx · b =
x when bx > 0
In particular, if b = − 1 and x < 0 , then d dx (ln (−x)) =^1 x when x < 0.
Thus, d dx (ln^ |x|) =
x when^ |x| 6^ = 0, This gives (^) ∫ 1 x dx = ln |x| + C
and (^) ∫ u′^ (x) u (x) dx^ = ln^ |u^ (x)|^ +^ C Example: Find (^) ∫ ex^ + cos x ex^ + sin x dx.
Example: Find (^) ∫ √ sec^ x ln (sec x + tan x)
dx
Familiar algebraic properties of ln x
Sketch of proof: Let u (x) = bxr, then
d dx ln u (x) = u′^ (x) u (x)
brxr−^1 bxr^
r x
∫ (^) y
1
d dx ln u (x) dy =
∫ (^) y
1
r x dy = r ln y ln u (y) − ln u (1) = r ln y ln (bxr) − ln b = r ln x ln (bxr) = ln b + r ln x The graph and range of ln x
- ln x is an increasing function because
- the graph of ln x is concave down because
The graph of y = ln−^1 x = ex^ is the graph of ln x reflected across the line y = x.
Example: Find d dx eln(sin^ x+1) Example: Find d dx ln(ex^2 −3 cos 2x) The Derivative and Integral of ex
(ex)′^ = ex
Why? Moreover, we have (^) ∫ exdx = ex^ + C. If u is any differentiable function of x, then by the Chain rule, d dx eu^ = eu^ du dx Thus, (^) ∫ eudu = eu^ + C Example: Suppose f (x) = e−3 sin (^2) (2x (^2) +1)
