# 6.7 Integrals, exponential functions, and logarithms

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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:

$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C,\phantom{\rule{0.2em}{0ex}}n\ne \text{−}1.$

Clearly, this does not work when $n=-1,$ as it would force us to divide by zero. So, what do we do with $\int \frac{1}{x}dx?$ Recall from the Fundamental Theorem of Calculus that ${\int }_{1}^{x}\frac{1}{t}dt$ is an antiderivative of $1\text{/}x.$ Therefore, we can make the following definition.

## Definition

For $x>0,$ define the natural logarithm function by

$\text{ln}\phantom{\rule{0.2em}{0ex}}x={\int }_{1}^{x}\frac{1}{t}dt.$

For $x>1,$ this is just the area under the curve $y=1\text{/}t$ from $1$ to $x.$ For $x<1,$ we have ${\int }_{1}^{x}\frac{1}{t}dt=\text{−}{\int }_{x}^{1}\frac{1}{t}dt,$ so in this case it is the negative of the area under the curve from $x\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}1$ (see the following figure).

Notice that $\text{ln}\phantom{\rule{0.2em}{0ex}}1=0.$ Furthermore, the function $y=1\text{/}t>0$ for $x>0.$ Therefore, by the properties of integrals, it is clear that $\text{ln}\phantom{\rule{0.2em}{0ex}}x$ 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

$\frac{d}{dx}\text{ln}\phantom{\rule{0.2em}{0ex}}x=\frac{1}{x}.$

## Corollary to the derivative of the natural logarithm

The function $\text{ln}\phantom{\rule{0.2em}{0ex}}x$ is differentiable; therefore, it is continuous.

A graph of $\text{ln}\phantom{\rule{0.2em}{0ex}}x$ is shown in [link] . Notice that it is continuous throughout its domain of $\left(0,\infty \right).$

what is function?
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
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)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
why n does not equal -1
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
y=800
800
Bg
how do u factor the numerator?
Nonsense, you factor numbers
Antonio
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
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
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
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
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)
Antonio
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
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
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
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
2x^3+6xy-4y^2)^2 solve this
femi
moe
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
what is a function? f(x)
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
is x=2 a function?
The
What is limit
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The