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  • Write the definition of the natural logarithm as an integral.
  • Recognize the derivative of the natural logarithm.
  • Integrate functions involving the natural logarithmic function.
  • Define the number e through an integral.
  • Recognize the derivative and integral of the exponential function.
  • Prove properties of logarithms and exponential functions using integrals.
  • Express general logarithmic and exponential functions in terms of natural logarithms and exponentials.

We already examined exponential functions and logarithms in earlier chapters. However, we glossed over some key details in the previous discussions. For example, we did not study how to treat exponential functions with exponents that are irrational. The definition of the number e is another area where the previous development was somewhat incomplete. We now have the tools to deal with these concepts in a more mathematically rigorous way, and we do so in this section.

For purposes of this section, assume we have not yet defined the natural logarithm, the number e , or any of the integration and differentiation formulas associated with these functions. By the end of the section, we will have studied these concepts in a mathematically rigorous way (and we will see they are consistent with the concepts we learned earlier).

We begin the section by defining the natural logarithm in terms of an integral. This definition forms the foundation for the section. From this definition, we derive differentiation formulas, define the number e , and expand these concepts to logarithms and exponential functions of any base.

The natural logarithm as an integral

Recall the power rule for integrals:

x n d x = x n + 1 n + 1 + C , n 1 .

Clearly, this does not work when n = −1 , as it would force us to divide by zero. So, what do we do with 1 x d x ? Recall from the Fundamental Theorem of Calculus that 1 x 1 t d t is an antiderivative of 1 / x . Therefore, we can make the following definition.


For x > 0 , define the natural logarithm function by

ln x = 1 x 1 t d t .

For x > 1 , this is just the area under the curve y = 1 / t from 1 to x . For x < 1 , we have 1 x 1 t d t = x 1 1 t d t , so in this case it is the negative of the area under the curve from x to 1 (see the following figure).

This figure has two graphs. The first is the curve y=1/t. It is decreasing and in the first quadrant. Under the curve is a shaded area. The area is bounded to the left at x=1. The area is labeled “area=lnx”. The second graph is the same curve y=1/t. It has shaded area under the curve bounded to the right by x=1. It is labeled “area=-lnx”.
(a) When x > 1 , the natural logarithm is the area under the curve y = 1 / t from 1 to x . (b) When x < 1 , the natural logarithm is the negative of the area under the curve from x to 1 .

Notice that ln 1 = 0 . Furthermore, the function y = 1 / t > 0 for x > 0 . Therefore, by the properties of integrals, it is clear that ln x is increasing for x > 0 .

Properties of the natural logarithm

Because of the way we defined the natural logarithm, the following differentiation formula falls out immediately as a result of to the Fundamental Theorem of Calculus.

Derivative of the natural logarithm

For x > 0 , the derivative of the natural logarithm is given by

d d x ln x = 1 x .

Corollary to the derivative of the natural logarithm

The function ln x is differentiable; therefore, it is continuous.

A graph of ln x is shown in [link] . Notice that it is continuous throughout its domain of ( 0 , ) .

This figure is a graph. It is an increasing curve labeled f(x)=lnx. The curve is increasing with the y-axis as an asymptote. The curve intersects the x-axis at x=1.
The graph of f ( x ) = ln x shows that it is a continuous function.

Questions & Answers

what is function?
Ryan Reply
A set of points in which every x value (domain) corresponds to exactly one y value (range)
what is lim (x,y)~(0,0) (x/y)
NIKI Reply
limited of x,y at 0,0 is nt defined
But using L'Hopitals rule is x=1 is defined
Could U explain better boss?
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
can we apply l hospitals rule for function of two variables
why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
I agree with Andrew
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
i mean can we replace the roles of x and y and call x=2 as function
if x =y and x = 800 what is y
Joys Reply
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The problem is the question, is not a problem where it is, but what it is
I think you should first know the basics man: PS
Yes, what factorization is
Antonio bro is x=2 a function?
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
Why y, if domain its usually defined as x, bro, so you creates confusion
Its f(x) =y=2 for every x
Yes but he said could you put x = 2 as a function you put y = 2 as a function
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
The said x=2 and that 2 is y
that 2 is not y, y is a variable 2 is a constant
So 2 is defined as f(x) =2
No y its constant =2
what variable does that function define
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Sorry x=2
And you are right, but os not a function of x, its a function of y
As function of x is meaningless, is not a finction
yeah you mean what I said in my first post, smh
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
OK you can call this "function" on a set {2}, but its a single value function, a constant
well as long as you got there eventually
2x^3+6xy-4y^2)^2 solve this
follow algebraic method. look under factoring numerator from Khan academy
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
It's an integral easy
V=1/3 h π (R^2+r2+ r*R(
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
is x=2 a function?
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
What is limit
MaHeSh Reply
it's the value a function will take while approaching a particular value
don ger it
what is a limit?
it is the value the function approaches as the input approaches that value.
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
is x=2 a function?

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