1.5 Exponential and logarithmic functions  (Page 2/17)

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 $\mathbit{\text{x}}$ $1.4$ $1.41$ $1.414$ $1.4142$ $1.41421$ $1.414213$ ${\mathbf{2}}^{\mathbit{\text{x}}}$ $2.639$ $2.65737$ $2.66475$ $2.665119$ $2.665138$ $2.665143$

Bacterial growth

Suppose a particular population of bacteria is known to double in size every $4$ hours. If a culture starts with $1000$ bacteria, the number of bacteria after $4$ hours is $n\left(4\right)=1000·2.$ The number of bacteria after $8$ hours is $n\left(8\right)=n\left(4\right)·2=1000·{2}^{2}.$ In general, the number of bacteria after $4m$ hours is $n\left(4m\right)=1000·{2}^{m}.$ Letting $t=4m,$ we see that the number of bacteria after $t$ hours is $n\left(t\right)=1000·{2}^{t\text{/}4}.$ Find the number of bacteria after $6$ hours, $10$ hours, and $24$ hours.

The number of bacteria after 6 hours is given by $n\left(6\right)=1000·{2}^{6\text{/}4}\approx 2828$ bacteria. The number of bacteria after $10$ hours is given by $n\left(10\right)=1000·{2}^{10\text{/}4}\approx 5657$ bacteria. The number of bacteria after $24$ hours is given by $n\left(24\right)=1000·{2}^{6}=64,000$ bacteria.

Given the exponential function $f\left(x\right)=100·{3}^{x\text{/}2},$ evaluate $f\left(4\right)$ and $f\left(10\right).$

$f\left(4\right)=900;f\left(10\right)=24,300.$

Go to World Population Balance for another example of exponential population growth.

Graphing exponential functions

For any base $b>0,b\ne 1,$ the exponential function $f\left(x\right)={b}^{x}$ is defined for all real numbers $x$ and ${b}^{x}>0.$ Therefore, the domain of $f\left(x\right)={b}^{x}$ is $\left(\text{−}\infty ,\infty \right)$ and the range is $\left(0,\infty \right).$ To graph ${b}^{x},$ we note that for $b>1,{b}^{x}$ is increasing on $\left(\text{−}\infty ,\infty \right)$ and ${b}^{x}\to \infty$ as $x\to \infty ,$ whereas ${b}^{x}\to 0$ as $x\to \text{−}\infty .$ On the other hand, if $0 is decreasing on $\left(\text{−}\infty ,\infty \right)$ and ${b}^{x}\to 0$ as $x\to \infty$ whereas ${b}^{x}\to \infty$ as $x\to \text{−}\infty$ ( [link] ).

Visit this site for more exploration of the graphs of exponential functions.

Note that exponential functions satisfy the general laws of exponents. To remind you of these laws, we state them as rules.

Rule: laws of exponents

For any constants $a>0,b>0,$ and for all x and y ,

1. ${b}^{x}·{b}^{y}={b}^{x+y}$
2. $\frac{{b}^{x}}{{b}^{y}}={b}^{x-y}$
3. ${\left({b}^{x}\right)}^{y}={b}^{xy}$
4. ${\left(ab\right)}^{x}={a}^{x}{b}^{x}$
5. $\frac{{a}^{x}}{{b}^{x}}={\left(\frac{a}{b}\right)}^{x}$

Using the laws of exponents

Use the laws of exponents to simplify each of the following expressions.

1. $\frac{{\left(2{x}^{2\text{/}3}\right)}^{3}}{{\left(4{x}^{-1\text{/}3}\right)}^{2}}$
2. $\frac{{\left({x}^{3}{y}^{-1}\right)}^{2}}{{\left(x{y}^{2}\right)}^{-2}}$
1. We can simplify as follows:
$\frac{{\left(2{x}^{2\text{/}3}\right)}^{3}}{{\left(4{x}^{-1\text{/}3}\right)}^{2}}=\frac{{2}^{3}{\left({x}^{2\text{/}3}\right)}^{3}}{{4}^{2}{\left({x}^{-1\text{/}3}\right)}^{2}}=\frac{8{x}^{2}}{16{x}^{-2\text{/}3}}=\frac{{x}^{2}{x}^{2\text{/}3}}{2}=\frac{{x}^{8\text{/}3}}{2}.$
2. We can simplify as follows:
$\frac{{\left({x}^{3}{y}^{-1}\right)}^{2}}{{\left(x{y}^{2}\right)}^{-2}}=\frac{{\left({x}^{3}\right)}^{2}{\left({y}^{-1}\right)}^{2}}{{x}^{-2}{\left({y}^{2}\right)}^{-2}}=\frac{{x}^{6}{y}^{-2}}{{x}^{-2}{y}^{-4}}={x}^{6}{x}^{2}{y}^{-2}{y}^{4}={x}^{8}{y}^{2}.$

Use the laws of exponents to simplify $\left(6{x}^{-3}{y}^{2}\right)\text{/}\left(12{x}^{-4}{y}^{5}\right).$

$x\text{/}\left(2{y}^{3}\right)$

The number e

A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from compounding interest in a savings account. Suppose a person invests $P$ dollars in a savings account with an annual interest rate $r,$ compounded annually. The amount of money after 1 year is

$A\left(1\right)=P+rP=P\left(1+r\right).$

The amount of money after $2$ years is

$A\left(2\right)=A\left(1\right)+rA\left(1\right)=P\left(1+r\right)+rP\left(1+r\right)=P{\left(1+r\right)}^{2}.$

More generally, the amount after $t$ years is

$A\left(t\right)=P{\left(1+r\right)}^{t}.$

If the money is compounded 2 times per year, the amount of money after half a year is

$A\left(\frac{1}{2}\right)=P+\left(\frac{r}{2}\right)P=P\left(1+\left(\frac{r}{2}\right)\right).$

The amount of money after $1$ year is

$A\left(1\right)=A\left(\frac{1}{2}\right)+\left(\frac{r}{2}\right)A\left(\frac{1}{2}\right)=P\left(1+\frac{r}{2}\right)+\frac{r}{2}\left(P\left(1+\frac{r}{2}\right)\right)=P{\left(1+\frac{r}{2}\right)}^{2}.$

After $t$ years, the amount of money in the account is

$A\left(t\right)=P{\left(1+\frac{r}{2}\right)}^{2t}.$

More generally, if the money is compounded $n$ times per year, the amount of money in the account after $t$ years is given by the function

What is a independent variable
a variable that does not depend on another.
Andrew
solve number one step by step
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
Hasnain
how to prove 1-sinx/cos x= cos x/-1+sin x?
1-sin x/cos x= cos x/-1+sin x
Rochel
how to prove 1-sun x/cos x= cos x / -1+sin x?
Rochel
how to prove tan^2 x=csc^2 x tan^2 x-1?
divide by tan^2 x giving 1=csc^2 x -1/tan^2 x, rewrite as: 1=1/sin^2 x -cos^2 x/sin^2 x, multiply by sin^2 x giving: sin^2 x=1-cos^2x. rewrite as the familiar sin^2 x + cos^2x=1 QED
Barnabas
how to prove sin x - sin x cos^2 x=sin^3x?
sin x - sin x cos^2 x sin x (1-cos^2 x) note the identity:sin^2 x + cos^2 x = 1 thus, sin^2 x = 1 - cos^2 x now substitute this into the above: sin x (sin^2 x), now multiply, yielding: sin^3 x Q.E.D.
Andrew
take sin x common. you are left with 1-cos^2x which is sin^2x. multiply back sinx and you get sin^3x.
navin
Left side=sinx-sinx cos^2x =sinx-sinx(1+sin^2x) =sinx-sinx+sin^3x =sin^3x thats proved.
Alif
how to prove tan^2 x/tan^2 x+1= sin^2 x
Rochel
Salim
what is function.
what is polynomial
Nawaz
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Alif
a term/algebraic expression raised to a non-negative integer power and a multiple of co-efficient,,,,,, T^n where n is a non-negative,,,,, 4x^2
joe
An expression in which power of all the variables are whole number . such as 2x+3 5 is also a polynomial of degree 0 and can be written as 5x^0
Nawaz
what is hyperbolic function
find volume of solid about y axis and y=x^3, x=0,y=1
3 pi/5
vector
what is the power rule
Is a rule used to find a derivative. For example the derivative of y(x)= a(x)^n is y'(x)= a*n*x^n-1.
Timothy
how do i deal with infinity in limits?
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
to understand how to model one variable as a direct relationship to another variable
Andrew