# 1.5 Exponential and logarithmic functions

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• Identify the form of an exponential function.
• Explain the difference between the graphs of ${x}^{b}$ and ${b}^{x}.$
• Recognize the significance of the number $e.$
• Identify the form of a logarithmic function.
• Explain the relationship between exponential and logarithmic functions.
• Describe how to calculate a logarithm to a different base.
• Identify the hyperbolic functions, their graphs, and basic identities.

In this section we examine exponential and logarithmic functions. We use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number $e.$ We also define hyperbolic and inverse hyperbolic functions, which involve combinations of exponential and logarithmic functions. (Note that we present alternative definitions of exponential and logarithmic functions in the chapter Applications of Integrations , and prove that the functions have the same properties with either definition.)

## Exponential functions

Exponential functions arise in many applications. One common example is population growth .

For example, if a population starts with ${P}_{0}$ individuals and then grows at an annual rate of $2%,$ its population after 1 year is

$P\left(1\right)={P}_{0}+0.02{P}_{0}={P}_{0}\left(1+0.02\right)={P}_{0}\left(1.02\right).$

Its population after 2 years is

$P\left(2\right)=P\left(1\right)+0.02P\left(1\right)=P\left(1\right)\left(1.02\right)={P}_{0}{\left(1.02\right)}^{2}.$

In general, its population after $t$ years is

$P\left(t\right)={P}_{0}{\left(1.02\right)}^{t},$

which is an exponential function. More generally, any function of the form $f\left(x\right)={b}^{x},$ where $b>0,b\ne 1,$ is an exponential function with base     $b$ and exponent     x . Exponential functions have constant bases and variable exponents. Note that a function of the form $f\left(x\right)={x}^{b}$ for some constant $b$ is not an exponential function but a power function.

To see the difference between an exponential function and a power function, we compare the functions $y={x}^{2}$ and $y={2}^{x}.$ In [link] , we see that both ${2}^{x}$ and ${x}^{2}$ approach infinity as $x\to \infty .$ Eventually, however, ${2}^{x}$ becomes larger than ${x}^{2}$ and grows more rapidly as $x\to \infty .$ In the opposite direction, as $x\to \text{−}\infty ,{x}^{2}\to \infty ,$ whereas ${2}^{x}\to 0.$ The line $y=0$ is a horizontal asymptote for $y={2}^{x}.$

 $\mathbit{\text{x}}$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ ${\mathbit{\text{x}}}^{\mathbf{2}}$ $9$ $4$ $1$ $0$ $1$ $4$ $9$ $16$ $25$ $36$ ${\mathbf{2}}^{\mathbit{\text{x}}}$ $1\text{/}8$ $1\text{/}4$ $1\text{/}2$ $1$ $2$ $4$ $8$ $16$ $32$ $64$

In [link] , we graph both $y={x}^{2}$ and $y={2}^{x}$ to show how the graphs differ.

## Evaluating exponential functions

Recall the properties of exponents: If $x$ is a positive integer, then we define ${b}^{x}=b·b\cdots b$ (with $x$ factors of $b\right).$ If $x$ is a negative integer, then $x=\text{−}y$ for some positive integer $y,$ and we define ${b}^{x}={b}^{\text{−}y}=1\text{/}{b}^{y}.$ Also, ${b}^{0}$ is defined to be $1.$ If $x$ is a rational number, then $x=p\text{/}q,$ where $p$ and $q$ are integers and ${b}^{x}={b}^{p\text{/}q}=\sqrt[q]{{b}^{p}}.$ For example, ${9}^{3\text{/}2}=\sqrt{{9}^{3}}=27.$ However, how is ${b}^{x}$ defined if $x$ is an irrational number? For example, what do we mean by ${2}^{\sqrt{2}}?$ This is too complex a question for us to answer fully right now; however, we can make an approximation. In [link] , we list some rational numbers approaching $\sqrt{2},$ and the values of ${2}^{x}$ for each rational number $x$ are presented as well. We claim that if we choose rational numbers $x$ getting closer and closer to $\sqrt{2},$ the values of ${2}^{x}$ get closer and closer to some number $L.$ We define that number $L$ to be ${2}^{\sqrt{2}}.$

#### Questions & Answers

Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
mukul Reply
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
ade
show that lim f(x) + lim g(x)=m+l
BARNABAS Reply
list the basic elementary differentials
Chio Reply
Differentiation and integration
Okikiola Reply
yes
Damien
proper definition of derivative
Syed Reply
the maximum rate of change of one variable with respect to another variable
Amdad
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)
Inembo Reply
what is calculus?
BISWAJIT Reply
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?
Geoffrey Reply
what is x and how x=9.1 take?
Pravin Reply
what is f(x)
Inembo Reply
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
Adri
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).
Adri
I dont understand what you wanna say by (A' n B^c)^c'
Adri
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Adri
Ok so the set is formed by vectors and not numbers
Adri
A vector of length n
Adri
But you can make a set out of matrixes as well
Adri
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
Adri
High-school?
Adri
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
Adri
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
Adri
I would say 24
Adri
Offer both
Adri
Sorry 20
Adri
Actually you have 40 - 4 =36 who offer maths or physics or both.
Adri
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
Adri
56-36=20 who give both courses... I would say that
Adri
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
how do i evaluate integral of x^1/2 In x
ayo Reply
first you simplify the given expression, which gives (x^2/2). Then you now integrate the above simplified expression which finally gives( lnx^2).
Ahmad
by using integration product formula
Roha
find derivative f(x)=1/x
Mul Reply
-1/x^2, use the chain rule
Andrew
f(x)=x^3-2x
Mul
what is domin in this question
noman
all real numbers . except zero
Roha
please try to guide me how?
Meher
what do u want to ask
Roha
?
Roha
the domain of the function is all real number excluding zero, because the rational function 1/x is a representation of a fractional equation (precisely inverse function). As in elementary mathematics the concept of dividing by zero is nonexistence, so zero will not make the fractional statement
Mckenzie
a function's answer/range should not be in the form of 1/0 and there should be no imaginary no. say square root of any negative no. (-1)^1/2
Roha
domain means everywhere along the x axis. since this function is not discontinuous anywhere along the x axis, then the domain is said to be all values of x.
Andrew
Derivative of a function
Waqar
right andrew ... this function is only discontinuous at 0
Roha
of sorry, I didn't realize he was taking about the function 1/x ...I thought he was referring to the function x^3-2x.
Andrew
yep...it's 1/x...!!!
Roha
true and cannot be apart of the domain that makes up the relation of the graph y = 1/x. The value of the denominator of the rational function can never be zero, because the result of the output value (range value of the graph when x =0) is undefined.
Mckenzie
👍
Roha
Therefore, when x = 0 the image of the rational function does not exist at this domain value, but exist at all other x values (domain) that makes the equation functional, and the graph drawable.
Mckenzie
👍
Roha
Roha are u A Student
Lutf
yes
Roha
What is the first fundermental theory of Calculus?
ZIMBA Reply
do u mean fundamental theorem ?
Roha
I want simple integral
aparna Reply
for MSc chemistry... simple formulas of integration
aparna
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funny
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funny
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aparna
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aparna
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aparna
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funny
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aparna
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funny
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RIZWAN
I don't understand the formula
Adaeze Reply
who's formula
funny
which formula?
Roha
what is the advantages of mathematical economics
Mubarak

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