Calculus Of One Real Variable

7.2 
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1. The Natural Logarithm Function 
Consider the relation 2^{3} = 8. We know that 2 is
the cube root of 8 and that 8 is the cube of 2. What about 3? Well, in this
case,
it's the exponent to which 2 is raised to get 8. It's given the name of
“logarithm of base 2 of 8”. Why isn’t it called just the
“exponent to which 2 is raised to get 8”? Of course it can be called just that,
because it is just that. Now consider the relation
2^{x} = y. This relation
defines y
as an exponential function of base 2 of x.
Of course x
is the exponent to which 2 is raised to get
y. Now consider the
inverse function, ie, the function that defines x as the exponent to which 2 is raised to
get y: x =
exponenttowhich2israisedtoget( y) (what a long name
of a function!!). If we call that function the “exponent function of
base 2”, there'll be confusion between it and the exponential function of base
2: y =
2^{x}, which is
exactly its opposite (inverse).
So we don't call it the “exponent function of base 2”, but rather the
“logarithm function of base 2”. Thus in 2^{3} = 8, 3 is given
the name “logarithm of base 2 of 8”.
For the definition and discussion of inverse functions, see Section 3.4.

Fig. 1.1 Graph Of y
= ln
x
Is Reflection Of Graph Of Its Inverse y
= e^{x} In The Line 
reflection of the graph of its inverse y = e^{x} in the line y = x. It's sketched in Fig. 1.1.
If y
= ln
x, then x = e^{y}.
Thus y
is the exponent to which e
is raised to get x.
This means that, since y
= ln
x, the natural
logarithm of x
is the exponent to which e
is raised to get x.
For simplicity, we'll say “the exponent on e
to get x”.
Since e
is
the base which is raised to y
to get x,
the natural logarithm is the logarithm of base e.
Definition 1.1  The Natural Logarithm Function
The inverse function of the natural exponential function e^{x} is called the natural logarithm function and is denoted ln x, where “ln” is pronounced like the word “lawn”. So: So ln x = exponent on e to get x. 
Remark 1.1
In “... the exponent on e to get x”, the quantity appearing after “to
get”, x
in this case, isn't the function (output), it's the
variable (input).
Composition
Recall that the composition of a function and its inverse
is the identity function: f
( f ^{1}(x))
= x
and f ^{1}( f
(x))
= x.
So ln
e^{x}
= x
and e^{ln}^{ }^{x}
= x.
We can see them intuitively as follows. Of course ln e^{x}
is the exponent on e
to get e^{x}, which is e raised to the
exponent x,
thus that exponent must be x.
As for e^{ln}^{ }^{x},
it's e
raised to a number that's the exponent on e
to get x,
consequently
e raised to that
exponent must be x.
They can be proved formally as follows.
For ln e^{x}
= x.
Let y =
ln
e^{x}.
Then e^{x} = e^{y},
hence x =
y, it follows that ln
e^{x}
= x.
For e^{ln}^{ }^{x}
= x.
Let y =
e^{ln}^{ }^{x}.
Then ln
x = ln
y, hence x = y, it follows that e^{ln}^{ }^{x}
= x.
Composition
ln e^{x}
= x, for all x, 
Properties
So in y = ln x, it must be that x > 0. This can also be seen from the fact that x = e^{y} and e^{y} > 0 for all y.
b. We have:
ln 1 = (exponent on e
to get 1) = 0, since e^{0} = 1,
ln e = (exponent on e to get e) = 1, since e^{1} = e.
We can prove
these values as follows. Let a
= ln
1. Then e^{a} = 1 = e^{0}. So ln 1 = a = 0, where the last
equation is obtained
from the property that the function e^{x}
is onetoone. Now let b
= ln
e. Then e^{b}
= e =
e^{1}. So ln
e = b = 1.
Or, using the
definition of inverse functions: e^{0} = 1, thus 0 = ln
1, or ln
1 = 0; now e^{1} = e,
thus 1 = ln e,
or ln
e = 1. This of
course is simpler.
These values are also marked in Fig. 1.1, together with the values e^{0} = 1 and e^{1} = e.
Now let y = ln x, so that x = e^{y}.
If 0 < x < 1, then 0 < e^{y}
< 1, which requires that y
< 0, thus ln x < 0.
If x > 1, then e^{y}
> 1, which requires that y
> 0, thus ln x > 0.
Properties
Refer to Fig. 1.1. b. If 0 < x < 1 then ln
x
< 0, 
The Function L(x)
In Section
7.1 Part 2, we defined the function L(x) as the signed area
bounded by y =
1/x,
the xaxis,
and the yaxis.
Refer to
Figs. 1.2 and 1.3. The function A(x) is the area of the
colored region. The function L(x) was defined as
follows:
We also showed that the inverse of L(x) is e^{x}.
So the inverse of e^{x} is L(x).
Since the inverse of a function is unique, the natural
logarithm function ln x is the same function as L(x).
Now you see why we called that function L(x): we used the letter
“L”

Fig. 1.2 

Fig. 1.3 L(x) = –A(x) if 0 < x < 1. 
because we had the word “Logarithm” in mind.
Natural
Logarithm As Signed Area
Consider the curve y = 1/t for t > 0. See Figs. 1.2 and 1.3. For any point x > 0, let A(x) be the area of the plane region bounded by the curve y = 1/t, the taxis, the vertical line t = 1, and the vertical line t = x. Let the function L(x) be defined as follows: Then L(x) is the natural logarithm function. 
In fact, another approach to introduce the exponential and
logarithmic functions in calculus is to present the natural logarithm
first, defined exactly as we did L(x), called natural
logarithm instead of L,
then present the natural exponential as the inverse
of the natural logarithm, then present the general exponential and logarithm.
To show that the general exponential b^{x} is
differentiable and to establish its other properties, either L(x)
or the natural logarithm must be introduced before any
exponential, because L(x) or the natural
logarithm is needed to show that the number e
exists, which in turn leads to the fact
of the differentiability of b^{x}, and to prove its
other properties.
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2. Properties 
One of the properties of the natural logarithm that we're
going to state and prove is ln xy = ln x + ln
y. This relation
expresses the natural logarithm of the product xy of 2 numbers x and y
in terms of the natural logarithms of the original
numbers x
and y.
This property says that the natural logarithm of the product of 2 numbers
equals the sum of the natural
logarithms of the original numbers. Intuitively, ln x + ln
y is the sum of the
exponent on e
to get x,
say s,
and the exponent
on e
to get y,
say t,
so that ln x
+ ln
y = s + t. Now e^{s}^{+}^{t}
= e^{s}e^{t}
= xy,
which indicates that s
+ t
is the exponent on e
to get xy.
Thus s +
t = ln
xy. As a consequence, ln
xy = ln
x + ln
y.
Theorem 2.1 – Properties Of Natural Logarithm
For any x > 0, any y > 0, and any t we have: 
Proof
a. Let s = ln x and t = ln y, so that x = e^{s} and y
= e^{t}. Then ln xy
= ln e^{s}e^{t} = ln e^{s}^{+}^{t} = s
+ t = ln x + ln y.
b. ln
(1/x) + ln x = ln ((1/x)x)
= ln 1 = 0, where the first equation is obtained
by part c. So ln (1/x) =  ln x.
c. ln (x/y) = ln (x(1/y))
= ln x
+ ln (1/y) = ln x  ln y, using parts c and d.
d. Note
that if t
is a positive integer then, by repetitive applications of part c:
For t = 0, we have ln x^{0} = ln 1 = 0 = 0 ln x.
For any non0 real number t. Let y = ln x^{t}, so that x^{t} = e^{y}. Then x
= e^{y}^{/}^{t}, then y/t = ln x, then y = t
ln x.
Thus ln x^{t} =
t ln x.
Consequently, for any real number t we have ln x^{t} = t
ln x.
EOP
Corollary 2.1  Natural Logarithms Of Extended Products And Quotients
For any real p_{i}, any real q_{j}, any positive integer m, and any positive integer n we have: 
Proof
EOP
Example 2.1
Simplify the following expressions.
a. ln 4 + 3 ln 2 –
2 ln 8.
b. ln(x + 1)^{2} – ln(x^{3 }+ x^{2 }– 3x
– 3) + ln(x^{2} – 3).
Solution
EOS
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3. Differentiation 
The derivative of L(x) is 1/x. So the derivative
of ln
x is 1/x. Of course the proof
for the case of ln x uses the fact that ln
x is
the inverse of e^{x}. It's much shorter
and much simpler than the proof for the case of L(x) as carried out in Section
7.1 Part 2.
Theorem 3.1  Derivative Of
Natural Logarithm
For all x > 0 we have: 
Proof
Let y = ln x. Then x = e^{y}. Differentiating this equation implicitly with respect to x we have:
EOP
Derivative Of ln x
Corollary 3.1  Derivative Of
Natural Logarithm Of Absolute Value

Proof
It remains to prove that (d/dx) ln x = 1/x if x
< 0. For any x
< 0, using the chain rule we have:
EOP
Example 3.1
Find the following derivatives. Restate your answers in the form of indefinite integrals.
Solution
EOS
Remark 3.1
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4. Graph 
y = ln x.
Inflection Points. Since there's no change of concavity,
there's no inflection point.
Special Characteristic. ln e = 1.
The graph of y = ln x is sketched in Fig. 4.1.

Fig. 4.1 Graph Of y = ln x. 
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5. Slowness 
From Graph
Refer to Fig. 5.1. When x increases from the near right of 0 to
1, ln
x increases as much as
from near minus infinity to 1. This
corresponds to the fact that when x
increases as much as from near minus infinity to 0, e^{x}
increases only from the near above

Fig. 5.1 The natural exponential y = e^{x} increases toward
infinity very fast, so its 
of 0 to 1. But when x
is greater than 1 and increases, ln x increases very slowly. This corresponds
to the fact that when x
is
positive and increases, e^{x} increases very fast.
In fact, ln x
increases more and more slowly, as evidenced by the fact that its
graph becomes less and less steep.
From Table Of Values
Now let's reach for a calculator and calculate the
exponents on e
as displayed in the table in Fig. 5.2. Clearly as x increases
from 1 to 1,000,000, ln x increases only from 0 to 13.82. Very
slow indeed. Of course in the table we can find the exponents
on e
by trial and error. However it's faster to just calculate the value of ln
x to use as the
exponent on e
to get x,
as we
actually did for this table!! Of course ln x is the exponent on e to get x.
Fig. 5.2 As x
increases from 1 to 1,000,000, ln x increases 
From Inverse
Suppose y
= f (x). If x changing by 1 unit
implies y
changing by 3 units, then y
changing by 1 unit implies x
changing by 1/3
unit. If y
changes faster than x
does, then of course x
changes slower than y
does. Now the inverse function is x
= f^{
}^{1}( y).
Here again if y
changes faster than x
does, then of course x
changes slower than y
does. Conforming to the good old tradition,
we now use the letter x
for the variable and the letter y
for the function (for more detail see Section
3.4). So now for the
inverse function we have y
= f ^{1}(x).
Thus if the y
in y =
f (x) changes faster than
the x
there does, then the y
in y =
f ^{1}(x)
changes slower than the x
there does. In brief, if a function changes fast, then its inverse changes slowly.
We saw in Section 7.1 Part 9 that the natural exponential function e^{x} grows very fast toward infinity, as asserted by this limit:
Consequently, its inverse the natural logarithm function must be very slow in its increase toward infinity.
Theorem 5.1  Slowness Of Growth Of Natural Logarithm
a. ln x < x for all x > 0. (In Fig. 5.1, the entire graph of y = ln x is below that of y = x.) 
Proof
EOP
Remarks 5.1 (On Theorem 5.1)
a. For all x > 0, ln x grows slower than x does.
b. By part a, ln
x^{t}
grows slower than x^{t} does. If 0 < t < 1, then x^{t}/t > x^{t},
so ln
x^{t}
must grow slower than x^{t}/t does. If t > 1, then
x^{t}/t < x^{t},
so we see that not only does ln x^{t}
grow slower than x^{t} does, but even slower
than x^{t}/t, which is smaller than x^{t},
does.
Problems & Solutions 
1. Simplify the following expressions.
a. 4 ln
2 + ln
3 – ln
6.
b. ln(x
+ 1) + ln(x –
1) – ln(x^{2} – 1).
c. ln(1 + cos
x)
+ ln(1
– cos
x).
Solution
a. 4 ln 2 + ln 3 – ln 6 = ln 2^{4} + ln(3/6) = ln 16 + ln(1/2) = ln(16 x 1/2) = ln 8.
b. ln(x + 1) + ln(x – 1) – ln(x^{2} – 1) = ln((x + 1)(x – 1)/(x^{2} – 1)) = ln((x^{2} – 1)/(x^{2} – 1)) = ln 1 = 0.
c. ln(1 + cos x) + ln(1 – cos x) = ln((1 + cos x)(1 – cos x)) = ln(1 – cos^{2}x) = ln sin^{2 }x = 2 ln sin x.
2.
Differentiate the following functions, simplify your answers whenever
possible, and restate the results as indefinite
integrals.
a. y = ln((2x
+ 3)/(x^{2} + 4)), x > –3/2, so that (2x
+ 3)/(x^{2} + 4) > 0.
b. y
= ln
csc x
+ cot
x.
c. y =
ln^{2} ln x
(ie, ( ln
ln
x)^{2}).
Solution
3. Find:
Solution
a. f '(x) = A cos ln x + Ax (– sin ln x)(1/x) + B sin ln x + Bx (cos ln x)(1/x) = (A + B ) cos ln x + (B – A) sin ln x.
b. Part a
shows that if B = A, then we would get rid of sin
ln
x.
So let g(x)
= Ax cos ln
x
+ Ax sin ln
x.
Then g'(x)
=
2A cos
ln
x.
It follows that:
Solution
5. Prove that for any t > 0:
Note
can be written as:
Solution
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