The document discusses the natural logarithm function ln(x) and the natural exponential function exp(x). It begins by defining ln(x) as the area under the curve y=1/t from 1 to x, and noting that its derivative is 1/x. It then defines exp(x) as the inverse of ln(x). It is shown that for rational r, exp(r) = er, and this definition is extended to irrational r. The derivative of exp(x) is then shown to be exp(x) itself.

1. 48 | T r a n s c e n d e n t a l F u n c t i o n s
Chapter 3
Transcendental Functions
3.1 The Natural Logarithm Function
The power of calculus, both that of derivatives and integrals, has already been
amply demonstrated. Yet we have only scratched the surface of potential
applications. To dig deeper, we need to expnnd the class of functions with which we
can work. That is the object of this chapter.
We begin by asking you to notice a peculiar gap in our knowledge of
derivatives.
Is there a function whose derivative is
In other words, is there an antiderivative
The First Fundamental Theorem of Calculus states that the accumulation
function
is a function whose derivative is f(x), provided that f is continuous on an interval I
that contains a and x. In this sense, we can find an antiderivative of ally continuous
function. The existence of an antiderivative does not mean that the antiderivative can
be expressed in terms of functions that we have studied so far. In this chapter we
will introduce and study a number of new functions.
Definition 3.1 Natural Logarithm Function
The natural logarithm function, denoted by ln, is defined by
The domain of the natural logarithm function is the set of positive real numbers.

2. 49 | T r a n s c e n d e n t a l F u n c t i o n s
The diagrams in Figure 1 indicate the geometric meaning of in x, The natural
logarithm or natural log) function measures the area under the curve y = 1/t between
1 and x if x > 1 and the negative of the area if 0 < x < 1. The natural logarithm is an
accumulation function because it accumulates area under the curve
.
Figure 1
. Clearly, In x is well defined for
is not defined for
because this
definite integral does not exist over an interval that includes 0.
And what is the derivative of this new function? Just exactly what we want.
Derivative of the Natural Logarithm Function
From the First Fundamental Theorem of CaLculus, we have
This can be combined with the Chain Rule. If
differentiable, then
EXAMPLE 1 Find
SOLUTION Let
Then
and if
f
is

3. 50 | T r a n s c e n d e n t a l F u n c t i o n s
EXAMPLE 2 Find
SOLUTION
This problem makes sense, provided that
.
which s positive provided that
Thus, the domain of
EXAMPLE 3
is
Now
or
. On this domain,
Show that
SOLUTION
wo cases are to be considered.
and
, and so
We know that for every differentiation formula there is a corresponding integration
formula. Thus. Example 3 implies that
or with replacing x,
This fills the long-standing gap in the Power Rule:
which we had to exclude the exponent
EXAMPLE 4
Find
SOLUTION
Let
EXAMPLE 5
Evaluate
so
. Then
, from
.

4. 51 | T r a n s c e n d e n t a l F u n c t i o n s
SOLUTION
Let
so
. Then
Thus, by the Second Fundamental Theorem of Calculus,
For the above calculation to be valid,
must never be 0 on the interval
It is easy to see that this is true.
When the integrand is the quotient of two polynomials (that is, a rational
function) and the numerator is of equal or greater degree than the denominator,
always divide the denominator into the numerator first,
EXAMPLE 6
Find
SOLUTION
By long division
Hence,
Properties of the Natural Logarithm
The next theorem lists several important properties of the natural log function.

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Theorem 3.1
If
and
are positive numbers and r is any rational number, then
(i)
(ii)
(ii)
(iv)
Proof
(i)
(ii)
Since, for
and
it follows from the theorem about two functions with the same derivative that
To determine C let
Finally, let
(iii)
, obtaining In
.
Replace a by
in (ii) to obtain
Thus,
Applying (ii) again, we get
(iv)
Since, for
and
We get
Finally, let
. Thus,

6. 53 | T r a n s c e n d e n t a l F u n c t i o n s
Logarithmic Differentiation
The labor of differentiating expressions involving quotients, products, or powers can
often be substantially reduced by first applying the. natural logarithm function and
using its properties. This method, called logarithmic differentiation, is illustrated in
Example 8.
EXAMPLE 8
Differentiate
SOLUTION
First we take natural logarithms; then we differentiate implicitly with respect to x.
Thus,
Example 8 could have been clone directly, without first taking logarithms, and we
suggest you try it. You should be able to make the two answers agree.
Trigonometric Integrals
Some trigonometric integrals can be evaluated using the natural log function.
EXAMPLE 9
Evaluate
SOLUTION
Since
we can make the substitution
obtain
Similarly,
EXAMPLE 10
Evaluate
.
,
, to

7. 54 | T r a n s c e n d e n t a l F u n c t i o n s
SOLUTION
For this one we use the trig identity
Then
Exercises 3.1
In problems 1-4, find the indicated derivative
1.
2.
3.
4.
In problems 5-10 , find the integrals
5.
6.
7.
8.
9.
10.
.

8. 55 | T r a n s c e n d e n t a l F u n c t i o n s
3.2 Inverse Functions and Their Derivatives
In this section, we study the general problem of reversing (or inverting) a
function. Here is the idea.
A function f takes a number x from its domain D and assigns to it a single
value y from its range R. If we are lucky, as in the case of the two functions graphed
in Figures 2 and 3, we can reverse f; that is, for any given y in R, we can
unambiguously g back and find the x from which it came, This new function that
takes y and assigns x to it is denoted by
. Note that its domain is R and its range
is Di It is called the inverse off or simply f-inverse. Here we are using the superscript
in a new way. The symbol
oes not denote.
. , as you might have expected
We, and all mathematicians, use it to name the inverse function.
Figure 3
Figure 2
Sometimes, we can give a formula for
(see
Figure
2).
Similarly,
If
if
then
then
(Figure 3). In each caset we simply solve the equation that
determines f for x in terms of y. The result is
.
But life is more complicated than these two examples indicate. Not every
function can be reversed in an unambiguous way. Consider
for
example. For a given y there are two is that correspond to it (Figure 4). The function

9. 56 | T r a n s c e n d e n t a l F u n c t i o n s
is even worse, For each y there are infinitely many x's that
correspond to it (Figure 5), Such functions do not have inverses; at least, they do not
unless we somehow restrict the set of x-values, a subject we will take up later.
Figure 4
Figure 5
Existence of Inverse Functions
It would be nice to have a simple criterion for deciding whether a function f has an
inverse. One such criterion is that the function be one-to-one; that is,
implies
This is equivalent to the geometric condition that every horizontal
line meet the graph of y =
in at most one point. But, in a given situation, this
criterion, may be very hard to apply, since it demands that we have complete
knowledge of the graph. A more practical criterion that covers most examples that
arise in this book is that a function be strictly monotonic. By this we mean that it is
either increasing or decreasing on its domain.
Theorem 3.2
If
is strictly monotonic on its domain, then
Proof Let
and
monotonic,
implies
has an inverse.
be distinct numbers in the domain off, with
or
. Since
. Either way,
is
Thus,
which means that f is one-to-one and therefore has
an inverse.
This is a practical result, because we have an easy way of deciding whether a
differentiable function f is strictly monotonic. We simply examine the sign of

10. 57 | T r a n s c e n d e n t a l F u n c t i o n s
EXAMPLE 11 Show that
has an inverse.
SOLUTION
for all . Thus, f is increasing on the whole real line and so it
has an inverse there.
We do not claim that we can always give a formula for
. In the example
just considered, this would require that we be able. to solve
for
. There is a way of salvaging the notion of inverse for functions that do not have
inverses on their natural domain. We simply restrict the domain to a set on which
the graph is either increasing or decreasing. Thus, for
restrict the domain to
(
, we may
would also work). For
restrict the domain to the interval
, we
. Then both functions have inverses (see
Figure 5), and we can even give a formula for the first one :
Figure 6
If
and
has an inverse
then
also has an inverse, namely, . Thus, we may call
a pair of inverse functions. One function undoes (or reverses) what the other
did that is,
EXAMPLE 2
Show that
has an inverse, find a formula for
, and verify the
results in the box above,
SOLUTION
Since f is an increasing function, it has an inverse. To find
for , which gives
and
, we solve
Finally, note that

11. 58 | T r a n s c e n d e n t a l F u n c t i o n s
The Graph of
Suppose that f has an inverse. Then
Consequently,
and
determine the same (x, y) pairs and so
have identical graphs. However, it is conventional to use x as the domain vari able
for functions, so we now inquire about the graph of
(note that we have
in the roles of x and y). A little thought convinces us that to inter- change the roles of
x and v on a graph is to reflect the graph across the line
Thus the graph of
the line
.
is just the reflection of the graph of
across
(Figure 6).
Figure 6
A related matter is that of finding a formula for
To do it, we first find
and then replace y by x in the resulting formula. Thus, we propose the following
three-step process for finding
Step 1 : Solve the equation
Step 2 : Use
for
in terms of .
to name the resulting expression in .
Step 3 : Replace y by x to get the formula fo
Before trying the three-step process on a particular function f, you might think we
should first verify that f has an inverse. However, if we can actually carry out the

12. 59 | T r a n s c e n d e n t a l F u n c t i o n s
first step and get a single x for each y, then
this for
we get
does exist. (Note that when we try
, which immediately shows that
does not exist, unless, of course, we have restricted the domain to eliminate one of
the two signs,
or .)
EXAMPLE 3
Find a formula dor
if
SOLUTION
Here are the three steps for this example.
Step 1:
Step 2:
Step 3:
Derivatives of Inverse Functions
We conclude this section by investigating the relationship between the derivative of a
function and the derivative of its inverse. Consider first what happens to a line
when it is reflected across the line y = x. As the left half of Figure 7 makes clear,
reflected into a line
provided
; moreover, their respective slopes
. If
and
is
are related by
happens to be the tangent line to the graph of f at

13. 60 | T r a n s c e n d e n t a l F u n c t i o n s
the point (c, d), then
is the tangent line to the graph of
at the point (d, c) (see
the right half of Figure 7), We are led to the conclusion that
Figure 7
Pictures are sometimes deceptive, so we claim only to have made the following
result plausible, For a formal proof, see any advanced calculus book.
Theorem 3.3 Inverse Function Theorem
Let
be differentiable and strictly monotonic on an interval I, If
certain x in , then
range of
differentiable at the corresponding point
and
The conclusion to Theorem B is often written symbolically as
at a
in the

14. 61 | T r a n s c e n d e n t a l F u n c t i o n s
EXAMPLE 4
Let
as in Example 1. Find
SOLUTION
Even though we cannot find a forma for
corresponds to
in this case, we note that
, and,since
,
Exercises 3.2
In Problems 1 – 5, find a formula for
dan
and then verify that
.
1.
2.
3.
4.
5.
6. If
and
then f has an inverse (why?) Let
)
Find
(a)
(b)
(c)
3.3 The Natural Exponential Function
The graph of
was obtained at the end of Section 3.1 and
is reproduced in Figure 1.The natural logarithm function is differentiable (hence
continuous) and increasing on its domain
): its range is
. It
is, in fact, precisely the kind of function studied in Section 3.2, and therefore has an
inverse In-1 with domain
and range
that it is given a special name and a special symbol
. This function is so important

15. 62 | T r a n s c e n d e n t a l F u n c t i o n s
Figure 1
Definition 3. 2
The inverse of In is called the natural exponential function and is denoted by exp,
Thus,
It follows immediately from this definition that
1.
2.
for all
Since exp and In are inverse functions, the graph of
= In x reflected across the line
(Figure2)
But why the name exponentiaifunction? You will see.
Figure 2
is just the graph of y

16. 63 | T r a n s c e n d e n t a l F u n c t i o n s
Definition 3.3
The letter
denotes the unique positive real number such that In
.
Figure 3 illustrates this definition: the area under the graph of
and
is 1. Since In e = 1, it is also true that
between
. The number ,
like , is irrational. Its decimal expansion is known to thousands of places; the first
few digits are
Figure 3
Now we make a crucial observation, one that depends only on facts already
demonstrated: (1) above and Theorem 3.1 (i). If
is any rational number,
Let us emphasize the result. For rational r, exp r is identical with
What was
introduced in the most abstract way as the inverse of the natural logarithm, which
itself was defined by an integraL has turned out to be a simple power.
But what if r is irrational'? Here we remind you of a gap in all elementary
algebra books. Never are irrational powers defined in anything approaching a
rigorous manner. What is meant by
? You will have a hard time pinning that
number down., based on elementary algebra, But we must pin it down if we are to
talk of such things as
for all
. Guided by what we learned above, we simply define
(rational or irrational) by
Note that (1) and (2) at the beginning of this section now take the following form:

17. 64 | T r a n s c e n d e n t a l F u n c t i o n s
Theorem 3.4
Let
and
any real numbers. Then
and
Proof To prove the first, we write
=
The second fact is proved similarly,
The Derivative of
Since exp and ln are inverses, we know from Theorem 3.2 (ii) that
differentiable. To find a formula for
is
we could use that the Alternatively, let
so that
Now differentiate both sides with respect to . Using the Chain Rule, we obtain
Thus,
We have proved the remarkable fact that
Thus,
is a solution of the differential equation
differentiable, then the Chain Rule yields
EXAMPLE 5
is its own derivative: that is,
Find
If
is

18. 65 | T r a n s c e n d e n t a l F u n c t i o n s
SOLUTION
Using
we obtain
EXAMPLE 6 Find
SOLUTION
EXAMPLE 7
Let
. Find where
is increasing and decreasing and where it is concave
upward and downward. Also, identify all extreme values and points of inflection.
Then, sketch the graph of
.
SOLUTION
and
Keeping in mind that
for all x, we see that
and
. Thus,
for all
is decreasing on
and has its minimum value at
Also
of
for
is
for
of
and
and concave upward on
Since
and increasing on
for
so the graph
and has a point of inflection at
the line
asymptote. This information supports the graph in Figure 4.
is a horizontal

19. 66 | T r a n s c e n d e n t a l F u n c t i o n s
Figure 4
The derivative formula
, or, with
EXAMPLE 8
Let
EXAMPLE 9
Evaluate
SOLUTION
Let
replacing ,
Evaluate
SOLUTION
automatically yields the integral formula
EXAMPLE 10
Evaluate
so
so
Then
. Then

20. 67 | T r a n s c e n d e n t a l F u n c t i o n s
SOLUTION
Let
so
. Then
Thus, by the Second Fundamental Theorem of Calculus,
The last recall can be obtained directly with a calculator.
EXAMPLE 11
Evaluate
SOLUTION
Think of
Although the symbol
Let
, so
will largely supplant exp
Then
throughout the rest of this book,
exp occurs frequently in scientific writing, especially when the exponent
is
complicated. For example, in statistics, one often encounters the normal probability
density function, which is

21. 68 | T r a n s c e n d e n t a l F u n c t i o n s
Exercises 3.3
In Problem 1 – 5, find
1.
2.
3.
4.
5.
(Hint: Use implicit differentiation)
In Problem 6 – 10, find each integral
6.
7.
8.
9.
10.

22. 69 | T r a n s c e n d e n t a l F u n c t i o n s
3.4 General Exponential and Logarithmic Functions
We defined
, and and all other irrational powers of e in the previous section.
But what about
and similar irrational powers of other numbers? In fact,
we want to give meaning to
for
a rational number, then
and
any real number. Now, if
is
. But we also know that
This suggests the definition of the exponential function to the base .
Definition 3.4
For
and any real number ,
Of course, this definition will he appropriate only if the usual properties of exponents
are valid for it, a matter we take up shortly. To shore up our confidence in the
definition, we use it to calculate
(with a little help from our calculator):
Your calculator may give a result that differs slightly from 9. Calculators use
approximations for
and
, and they round to a fixed number of decimal places
(usually about 8)
Properties of
Theorem 3.5 Properties of Exponents
If
, and
and y are real numbers, then
Proof We will prove (ii) and (iii), leaving the others for you.

23. 70 | T r a n s c e n d e n t a l F u n c t i o n s
Theorem 3.5 Exponential Function Rules
Proof
The integral formula follows immediately from the derivative formula.
EXAMPLE 12
SOLUTION
EXAMPLE 13
SOLUTION
Find
We use the Chain Rule with
Find
if

24. 71 | T r a n s c e n d e n t a l F u n c t i o n s
EXAMPLE 14
SOLUTION
Find
Let
so
Then
The Function
Finally, we are ready to make a connection with the
algebra, We note that if
then
rithms that you studied in
is a decreasing function; if
, it is an increasing function, as you may check by considering the derivative.
In either case,
has an inverse. We call this inverse the logarithmic function to the
base . This is equivalent to the following definition.
Definition 3.6
Let
be a positive. number different from 1. Then
Historically, the most commonly used base was base 10, and the resulting logarithms
were called common logarithms. But in calculus and all of advanced mathematics,
the significant base is . Notice that
, being the inverse of
, is just
another synthol for In; that is,
We have come full circle (Figure 1), The. function
, which we introduced in
Section 6.1, has turned out to he an ordinary logarithm, but t a rather special base,
Now observe that if
, so that
, then
from which we conclude that
.

25. 72 | T r a n s c e n d e n t a l F u n c t i o n s
Also,
EXAMPLE 15
SOLUTION
If
find
Let
The Functions
.
and apply the Chain Rule.
and
We have just learned that
What about
? For
rational, we proved the Power Rule in Chapter 2, which
says that
Now ‘ve assert that this is true even if a is irrational. To see this, write
The corresponding rule for integrals also holds even if
Finally, we consider
for
is irrational.
a variable to a variable power. There is a formula
, but we do not recommend that you memorize it. Rather, we. suggest that
you learn two methods for finding it, as illustrated below.
EXAMPLE 16
If
find
by two different methods

26. 73 | T r a n s c e n d e n t a l F u n c t i o n s
SOLUTION
Method 1 We may write
Thus using the Chain Rule and the Product Rule,
Method 2 Recall the logarithmic differentiation technique from Section 3.1,
EXAMPLE 17
If
find
SOLUTION
EXAMPLE 18
If
find
SOLUTION We use logarithmic differentiation.
.

27. 74 | T r a n s c e n d e n t a l F u n c t i o n s
Exercises 3.4
In Problem 1 – 10, find the indicated derivative or integral.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.