# 6.7 Integrals, exponential functions, and logarithms  (Page 3/4)

 Page 3 / 4
$\text{exp}\left(\text{ln}\phantom{\rule{0.2em}{0ex}}x\right)=x\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}x>0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{ln}\left(\text{exp}\phantom{\rule{0.2em}{0ex}}x\right)=x\phantom{\rule{0.2em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}x.$

The following figure shows the graphs of $\text{exp}\phantom{\rule{0.2em}{0ex}}x$ and $\text{ln}\phantom{\rule{0.2em}{0ex}}x.$

We hypothesize that $\text{exp}\phantom{\rule{0.2em}{0ex}}x={e}^{x}.$ For rational values of $x,$ this is easy to show. If $x$ is rational, then we have $\text{ln}\left({e}^{x}\right)=x\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}e=x.$ Thus, when $x$ is rational, ${e}^{x}=\text{exp}\phantom{\rule{0.2em}{0ex}}x.$ For irrational values of $x,$ we simply define ${e}^{x}$ as the inverse function of $\text{ln}\phantom{\rule{0.2em}{0ex}}x.$

## Definition

For any real number $x,$ define $y={e}^{x}$ to be the number for which

$\text{ln}\phantom{\rule{0.2em}{0ex}}y=\text{ln}\left({e}^{x}\right)=x.$

Then we have ${e}^{x}=\text{exp}\left(x\right)$ for all $x,$ and thus

${e}^{\text{ln}\phantom{\rule{0.2em}{0ex}}x}=x\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}x>0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{ln}\left({e}^{x}\right)=x$

for all $x.$

## Properties of the exponential function

Since the exponential function was defined in terms of an inverse function, and not in terms of a power of $e,$ we must verify that the usual laws of exponents hold for the function ${e}^{x}.$

## Properties of the exponential function

If $p$ and $q$ are any real numbers and $r$ is a rational number, then

1. ${e}^{p}{e}^{q}={e}^{p+q}$
2. $\frac{{e}^{p}}{{e}^{q}}={e}^{p-q}$
3. ${\left({e}^{p}\right)}^{r}={e}^{pr}$

## Proof

Note that if $p$ and $q$ are rational, the properties hold. However, if $p$ or $q$ are irrational, we must apply the inverse function definition of ${e}^{x}$ and verify the properties. Only the first property is verified here; the other two are left to you. We have

$\text{ln}\left({e}^{p}{e}^{q}\right)=\text{ln}\left({e}^{p}\right)+\text{ln}\left({e}^{q}\right)=p+q=\text{ln}\left({e}^{p+q}\right).$

Since $\text{ln}\phantom{\rule{0.2em}{0ex}}x$ is one-to-one, then

${e}^{p}{e}^{q}={e}^{p+q}.$

As with part iv. of the logarithm properties, we can extend property iii. to irrational values of $r,$ and we do so by the end of the section.

We also want to verify the differentiation formula for the function $y={e}^{x}.$ To do this, we need to use implicit differentiation. Let $y={e}^{x}.$ Then

$\begin{array}{ccc}\hfill \text{ln}\phantom{\rule{0.2em}{0ex}}y& =\hfill & x\hfill \\ \hfill \frac{d}{dx}\text{ln}\phantom{\rule{0.2em}{0ex}}y& =\hfill & \frac{d}{dx}x\hfill \\ \hfill \frac{1}{y}\phantom{\rule{0.1em}{0ex}}\frac{dy}{dx}& =\hfill & 1\hfill \\ \hfill \frac{dy}{dx}& =\hfill & y.\hfill \end{array}$

Thus, we see

$\frac{d}{dx}{e}^{x}={e}^{x}$

as desired, which leads immediately to the integration formula

$\int {e}^{x}dx={e}^{x}+C.$

We apply these formulas in the following examples.

## Using properties of exponential functions

Evaluate the following derivatives:

1. $\frac{d}{dt}{e}^{3t}{e}^{{t}^{2}}$
2. $\frac{d}{dx}{e}^{3{x}^{2}}$

We apply the chain rule as necessary.

1. $\frac{d}{dt}{e}^{3t}{e}^{{t}^{2}}=\frac{d}{dt}{e}^{3t+{t}^{2}}={e}^{3t+{t}^{2}}\left(3+2t\right)$
2. $\frac{d}{dx}{e}^{3{x}^{2}}={e}^{3{x}^{2}}6x$

Evaluate the following derivatives:

1. $\frac{d}{dx}\left(\frac{{e}^{{x}^{2}}}{{e}^{5x}}\right)$
2. $\frac{d}{dt}{\left({e}^{2t}\right)}^{3}$
1. $\frac{d}{dx}\left(\frac{{e}^{{x}^{2}}}{{e}^{5x}}\right)={e}^{{x}^{2}-5x}\left(2x-5\right)$
2. $\frac{d}{dt}{\left({e}^{2t}\right)}^{3}=6{e}^{6t}$

## Using properties of exponential functions

Evaluate the following integral: $\int 2x{e}^{\text{−}{x}^{2}}dx.$

Using $u$ -substitution, let $u=\text{−}{x}^{2}.$ Then $du=-2x\phantom{\rule{0.2em}{0ex}}dx,$ and we have

$\int 2x{e}^{\text{−}{x}^{2}}dx=\text{−}\int {e}^{u}du=\text{−}{e}^{u}+C=\text{−}{e}^{\text{−}{x}^{2}}+C.$

Evaluate the following integral: $\int \frac{4}{{e}^{3x}}dx.$

$\int \frac{4}{{e}^{3x}}dx=-\frac{4}{3}{e}^{-3x}+C$

## General logarithmic and exponential functions

We close this section by looking at exponential functions and logarithms with bases other than $e.$ Exponential functions are functions of the form $f\left(x\right)={a}^{x}.$ Note that unless $a=e,$ we still do not have a mathematically rigorous definition of these functions for irrational exponents. Let’s rectify that here by defining the function $f\left(x\right)={a}^{x}$ in terms of the exponential function ${e}^{x}.$ We then examine logarithms with bases other than $e$ as inverse functions of exponential functions.

## Definition

For any $a>0,$ and for any real number $x,$ define $y={a}^{x}$ as follows:

$y={a}^{x}={e}^{x\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}a}.$

Now ${a}^{x}$ is defined rigorously for all values of x . This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of $r.$ It is straightforward to show that properties of exponents hold for general exponential functions defined in this way.

find the domain and range of f(x)= 4x-7/x²-6x+8
find the range of f(x)=(x+1)(x+4)
-1, -4
Marcia
That's domain. The range is [-9/4,+infinity)
Jacob
If you're using calculus to find the range, you have to find the extrema through the first derivative test and then substitute the x-value for the extrema back into the original equation.
Jacob
Good morning,,, how are you
d/dx{1/y - lny + X^3.Y^5}
How to identify domain and range
hello
Akpevwe
He,,
Harrieta
hi
Dr
hello
velocity
I only talk to girls
Dr
women are smart then guys
Dr
Smarter
sorry
Dr
Dr
:(
Shun
was up
Dr
hello
is it chatting app?.. I do not see any calculus here. lol
Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
show that lim f(x) + lim g(x)=m+l
list the basic elementary differentials
Differentiation and integration
yes
Damien
proper definition of derivative
the maximum rate of change of one variable with respect to another variable
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
what is calculus?
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
what is x and how x=9.1 take?
what is f(x)
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
I dont understand what you wanna say by (A' n B^c)^c'
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Ok so the set is formed by vectors and not numbers
A vector of length n
But you can make a set out of matrixes as well
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
High-school?
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
I would say 24
Offer both
Sorry 20
Actually you have 40 - 4 =36 who offer maths or physics or both.
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
56-36=20 who give both courses... I would say that
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie