Chapter 19. Exponentials and Natural Logarithms
Questions
Volume 3, Why Slopes and More Math.
What does an electronic calculator compute when the natural logarithm
button on it is pressed? The answer follows from a long chain of
mathematical concepts and reasoning. The definition below of the natural
logarithm in terms of area under a curve $y=\frac1s$ provides a preview
(or review) of notions employed in integral calculus --- the subject
which treats, in the first instance, the calculation of areas under
curves. \em This chapter assumes a previous acquaintance with logarithms,
powers and exponentials.
Electronic Calculators
Electronic calculators allow the illustration and an electronic,
pre-programmed confirmation of the basic relationships between
logarithms, exponentials and powers. The following computations can be
illustrated with an electronic calculator.
This represents or indicates the easy buttons-on-a-calculator approach
to the description/explanation of logarithms, exponentials and powers
together with the relationships between the calculations invoked by the
buttons.
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The logarithm of $x > 0$ to a base $a > 0$ is given by
$$\log_a(x) = \frac{\ln(x)}{\ln(a)}\cdot$$ For example on some
electronic calculators, $\log_{10}(6)$ is giving by pressing the
$6$ and then the $\log$ button (in some order). This should give
the same result as computing $\ln(6)\div\ln(10)$. (Exercise: check
this by pushing buttons.)
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Multiplying a number $a > 0$ by itself $n$ times gives $a^n$.
But the calculation of $\exp(n \ln(a))$ gives the same result. So
the original definition of $a^n$ for $a >0$ is consistent with
the more general definition given for real numbers $x$ by
$$a^x=\exp(x \ln(a)).$$ For examples, compute $5^2$ and
$\exp(2\ln(5))$ on an electronic calculator. Also compute the
following: $x=10^{0.2}$, \quad $y=\exp(0.2\ln(10))$, \quad $x^5$
and $\exp(10\ln(x))$.
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For $a > 0$, \[ \log_a(a^x)=x\] and for $v>0$, $u=\log_a(v)$
implies $a^u=v$. For examples compute $\log(10^3)-3$ and
$10^{\log(8)}$.
The special cases a = 10 and a = 2 are of interest most likely due to
the recent historical preference for decimal (base 10) arithmetic and due
to, still more recently, to the advent of computers with their binary (base
2) arithmetic. Also of interest is a third case a = e where e is the
so-called natural number. See below.
Definitions of logarithms and exponential functions are given in the next
two webpages to explain and derive the computational relationships
described above.
The Natural Logarithm
For real numbers, the following sections describe the area-under-a-curve
definition of the natural logarithm, and how this introduction of the
natural logarithm leads to the definition and properties of all
logarithms, exponentials and powers involving real numbers.
The presentation here is to show briefly the approach I would
like to see favored in schools. Working through the details of this
exposition in its present form could be a subject for discussion in a
high school math club. Understanding this section and the next demands
or provides a sound command of some mathematics beyond arithmetic.
The natural logarithm ln( a) for a > 0 can be introduced as the
(signed) area under the curve $y = \frac 1s $ from s = 1 to s = a.
Equivalently, it may be represented by the signed area under the curve
$u=\frac1v$ from v = 1 to v = a. This definition does not depend on the
labelling of the horizontal and vertical axes. See the next two diagrams.
In the next diagram, the area from s = 1 to s = a > 1 can be
approximated by slicing it into n vertical rectangles with the same base
size $\frac {a-1}n$, and then making this base size smaller by letting
$n\to\infty$ (that is get larger and larger).
The shorthand $n\to\infty$ should be read as n tends to (or goes to)
infinity. It is left as an exercise for advance students to write on
paper the Riemann sums whose limit is or should be the value L.
The sum of the area of the resulting rectangles approximates
to a single number L with greater and greater accuracy, more decimal places
say, as $n \to\infty$. This single limit gives what we call ln( a).
For a 鲁 1, the value of ln( a) is given by the
area from s = 1 to s = a under the curve $y = \frac 1s $. Here we take or
assume ln(1) = 0. It can be shown that ln( a) $\to$ 0 when when a
approaches 1 through values above or greater than 1.
The natural logarithm ln( b) of a number b when 0 < b < 1 is
defined next.
For 0 < b < 1, the value of ln( b) is given by (-1) times the area
under the curve $y = \frac 1s $ from s = b to s = 1.
The above two diagram illustrate the arithmetic or area-based definition
of the natural logarithm ln( a) or ln( b) in the two mutually exclusive
cases a > 1 and 0 < b < 1. These definitions imply that ln( x)
$\to$ 0 = ln(1) when x $\to$ 1 through values > 1.
Reading Guide. The rest of this section states and indicates the
proofs of two algebraic properties of the natural logarithm. The first
proof is easy. The second proof is cryptic - material for advanced
students. The next section briefly indicates the relationship between the
inverses of the logarithms and exponential functions - more material for
advance students. Consult another calculus or analysis text for the
missing details.
Proof of Property \( \ln(\frac1b) = -\ln(b) $ for b > 0.
We will show that \( 0 =\ln(\frac1b) = -\ln(b) $ when b > 0.
For this, first consider the case a > 1. In the following
diagram
By symmetry (or reflection across the line y = s), ln( a) = Area(
B)+Area( A). Therefore ln( a) = Area( B)+Area( C)
Here A is the rectangle with corners (0,1) and (\(\frac1a$, 1) while C
is the rectangle with corners (\(\frac1a$,0) and (1,1)
Now by definition -ln([\(\frac1a$]) =
Area( B)+Area( C).
Therefore -ln([\(\frac1a$]) = ln( a).
This in turn implies ln([\(\frac1a$])+ln( a) = 0.whenever a > 1.
Finally, we conclude ln([\(\frac1b$])+ln( b) = 0 whenever b >
0. This follows by putting a = b if b 鲁 1 and
by putting a = [\(\frac1b$] if 0 < b < 1.) The latter is
equivalent to the property ln([\(\frac1b$]) = -ln( b) which we wanted to show.
Fundamental Property of Logarithms
Next we may derive the fundamental property of logarithms, that is
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ln( ab) = ln( b) +ln( a).
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(This holds when a = 1 and b > 0 since ln(1) = 0 by definition.)
We will now consider the case where a > 1 and b > 0. For this it
suffices to reconsider how the number ln( a) is computed. Two ways to show
this are indicated next.
Sketch of A First Demonstration
1. Divide the interval [1, a] on the s-axis into n 鲁 1 segments using the end points s i = 1+
i路[( a-1)/( n)] where 1 拢 i 拢 n. Each segment has
length [( a-1)/( n)].
2. On each segment [ s i, s i+1] construct
a rectangle whose top just touches the curve y = \(\frac1s$ at y =
\(\frac1{s_i}$. The sum S n of the areas
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A j = y j路( s i+1- s i) = y i路
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a-1
n
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of these rectangles provides an approximation to ln( a) which we assume
becomes more accurate as n is made larger.
3. Now the rectangle with base [ s i, s
i+1] and height \(\frac1{s_i}$ has the same area as the
rectangle with base [ bs i, bs i+1] and height
\(\frac1{bs_i}$. But the rectangles with base segments [ bs
i, bs i+1] and height \(\frac1{bs_i}$
approximate the area S ba under the curve y = \(\frac1s$
from s = b to s = ba. So taking the limit as n -> 楼 suggests S
ba = ln( a).
4. Drawing a graph suggests or implies S ba = ln( ab)
-ln( b). Therefore ln( a) = S ab
= ln( ab)-ln( b) as well. So we are done in
the first case where a > 1 and b > 0. That is, the area S
ba under the curve y = \(\frac1{s}$ from s = b to s = ba
equals the area under the curve y = \(\frac1{s}$ from s = 1 to s = ba
minus the areas from s = 1 to s = b.
Now the fundamental property of logarithms, that is ln( ab) = ln( b)
+ln( a) holds whenever at least one of the factors a and b is greater
than 1 (since addition and multiplication of real numbers is
commutative.) Now observe for c > 0 that 0 = ln(1) = ln(
\(\frac1{c}$ ) 路 c) = ln(\(\frac1{s }$)+ln( c) since c or its
reciprocal must be 鲁 1. Hence ln( c) =
- ln([\(\frac1c$). This was shown before
with the aid of some diagrams. The latter equality prepares us to treat
the sole remaining case where both numbers a and b are between 0 and 1.
In this case,
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-ln(
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1
a
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) + -ln(
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1
b
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) = ln( a)+ln( b)
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as required. Therefore ln( ab) = ln( a)+ln( b) holds whenever a
and b are both positive.
This indicates a simple demonstration of the fundamental
property for the natural logarithm ln( x) for x > 0. The sketch of
an alternative proof follows.
Sketch of a Second Demonstration. For a > 0, put G( x) = ln(
ax). Then value of G( x) is given by the (signed) area from s = 1 to s =
ax under the curve y = \(\frac1{s}$. Observe G(1) = ln( a). The area of
region D in the following diagram equals $G(x+\Delta x)-G(x)$.
The height of the region $D$ is approximately $\frac 1{ax}$ and its
length is precisely $a(x+\Delta x) - ax=a\Delta x$. Therefore \[
G(x+\Delta x)-G(x) \approx\mbox{Area}(D)=\frac 1{ax}\cdot a\Delta
x=\frac1x\cdot \Delta x \] This suggests that \[ G'(x)=\lim_{\Delta x\to
0}\frac{G(x+\Delta x)-G(x)}{\Delta x} =\frac1x \] Similarly $F(x)=\ln(x)$
implies that $F'(x)=\frac1x$. This implies by the \em Constant Difference
Theorem \rm that $$\ln(ax)-\ln(x)=G(x)-F(x)=d$$ is constant. To evaluate
the constant, observe that $$d=G(1)-F(1)=\ln(a)-\ln(1)=\ln(a)$$ since
$\ln(1)=0$. Thus we conclude $\ln(ax)-\ln(x)=\ln(a)$ or equivalently
\[\ln(ax)=\ln(a)+\ln(x) \] as required.
Logarithms To Base a > 0
The logarithm to base $a>0$ is given by
$\log_a(x)=\frac{\ln(x)}{\ln(a)}$ when $a\ne1$. The property
$\ln(ax)=\ln(a)+\ln(x)$ now implies $\log_c(ab) = \log_c(b) +\log_c(a)
$ holds when $a$, $b$ and $c$ are all positive real numbers with $c\ne
1$. The proof is a simple algebraic exercise. Further note that
$\ln(e)=1$ implies $\log_e(x)=\ln(x)$.
Inverse Functions and Exponentials
A well-known theory briefly described
The above geometric definition implies ln(1) = 0. It also implies that
ln(2) > 0.5 Note the rectangle of height 0.5 with base segment [1,2]
has area 0.5. It also lies strictly beneath the curve u = \(\frac1{v}$
where 1 拢 v 拢 2.
Now mathematical induction implies ln(2 n) = n ln(2) >
n/2 (since 2 n+1 = 2 n 路2).
Now ln(4) = ln(2)+ln(2) > 0.5+0.5 = 1. The continuity of ln( x) can
be shown directly. It is also a consequence of the differentiability of
the function ln(x) The continuity of ln( x) between x = 1 and x = 4
implies by the Intermediate Value Theorem there is at least one number
e such that ln( e) = 1. The number value y of the exponential function
exp( x) can now be defined as the unique number y satisfying the
equation ln( y) = x.
This definition of exp( x) leads to the property
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exp( x1) 路exp( x2) = exp(
x1+ x2)
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The stage is now set for derivation of the algebraic properties
of the exponential expressions a b and the logarithm log
a( b). That can include a discussion of roots and powers for
positive numbers.
Note that the number e is called the natural number.
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The infinite decimal expansion of e begins with 2.718281828
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the digits 1828 appear twice in this otherwise non-repeating
decimal expansion.
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The number e is irrational. The proof of that e is not rational, is
another intellectual mortgage.
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