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Values of 2 x For a list of rational numbers approximating 2
x 1.4 1.41 1.414 1.4142 1.41421 1.414213
2 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 ( 4 ) = 1000 · 2 . The number of bacteria after 8 hours is n ( 8 ) = n ( 4 ) · 2 = 1000 · 2 2 . In general, the number of bacteria after 4 m hours is n ( 4 m ) = 1000 · 2 m . Letting t = 4 m , we see that the number of bacteria after t hours is n ( t ) = 1000 · 2 t / 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 ( 6 ) = 1000 · 2 6 / 4 2828 bacteria. The number of bacteria after 10 hours is given by n ( 10 ) = 1000 · 2 10 / 4 5657 bacteria. The number of bacteria after 24 hours is given by n ( 24 ) = 1000 · 2 6 = 64,000 bacteria.

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Given the exponential function f ( x ) = 100 · 3 x / 2 , evaluate f ( 4 ) and f ( 10 ) .

f ( 4 ) = 900 ; f ( 10 ) = 24 , 300 .

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

Graphing exponential functions

For any base b > 0 , b 1 , the exponential function f ( x ) = b x is defined for all real numbers x and b x > 0 . Therefore, the domain of f ( x ) = b x is ( , ) and the range is ( 0 , ) . To graph b x , we note that for b > 1 , b x is increasing on ( , ) and b x as x , whereas b x 0 as x . On the other hand, if 0 < b < 1 , f ( x ) = b x is decreasing on ( , ) and b x 0 as x whereas b x as x ( [link] ).

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of four functions. The first function is “f(x) = 2 to the power of x”, an increasing curved function, which starts slightly above the x axis and begins increasing. The second function is “f(x) = 4 to the power of x”, an increasing curved function, which starts slightly above the x axis and begins increasing rapidly, more rapidly than the first function. The third function is “f(x) = (1/2) to the power of x”, a decreasing curved function with decreases until it gets close to the x axis without touching it. The third function is “f(x) = (1/4) to the power of x”, a decreasing curved function with decreases until it gets close to the x axis without touching it. It decrases at a faster rate than the third function.
If b > 1 , then b x is increasing on ( , ) . If 0 < b < 1 , then b x is decreasing on ( , ) .

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. b x b y = b x y
  3. ( b x ) y = b x y
  4. ( a b ) x = a x b x
  5. a x b x = ( a b ) x

Using the laws of exponents

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

  1. ( 2 x 2 / 3 ) 3 ( 4 x −1 / 3 ) 2
  2. ( x 3 y −1 ) 2 ( x y 2 ) −2
  1. We can simplify as follows:
    ( 2 x 2 / 3 ) 3 ( 4 x −1 / 3 ) 2 = 2 3 ( x 2 / 3 ) 3 4 2 ( x −1 / 3 ) 2 = 8 x 2 16 x −2 / 3 = x 2 x 2 / 3 2 = x 8 / 3 2 .
  2. We can simplify as follows:
    ( x 3 y −1 ) 2 ( x y 2 ) −2 = ( x 3 ) 2 ( y −1 ) 2 x −2 ( y 2 ) −2 = x 6 y −2 x −2 y −4 = x 6 x 2 y −2 y 4 = x 8 y 2 .
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Use the laws of exponents to simplify ( 6 x −3 y 2 ) / ( 12 x −4 y 5 ) .

x / ( 2 y 3 )

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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 ( 1 ) = P + r P = P ( 1 + r ) .

The amount of money after 2 years is

A ( 2 ) = A ( 1 ) + r A ( 1 ) = P ( 1 + r ) + r P ( 1 + r ) = P ( 1 + r ) 2 .

More generally, the amount after t years is

A ( t ) = P ( 1 + r ) t .

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

A ( 1 2 ) = P + ( r 2 ) P = P ( 1 + ( r 2 ) ) .

The amount of money after 1 year is

A ( 1 ) = A ( 1 2 ) + ( r 2 ) A ( 1 2 ) = P ( 1 + r 2 ) + r 2 ( P ( 1 + r 2 ) ) = P ( 1 + r 2 ) 2 .

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

A ( t ) = P ( 1 + r 2 ) 2 t .

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

Questions & Answers

how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
for the function f(x)={x^2-7x+104 x<=7 7x+55 x>7' does limx7 f(x) exist?
find dy÷dx (y^2+2 sec)^2=4(x+1)^2
Rana Reply
Integral of e^x/(1+e^2x)tan^-1 (e^x)
naveen Reply
why might we use the shell method instead of slicing
Madni Reply
fg[[(45)]]²+45⅓x²=100
albert Reply
find the values of c such that the graph of f(x)=x^4+2x^3+cx^2+2x+2
Ramya Reply
anyone to explain some basic in calculus
Adegoke Reply
A conical container of radius 10 ft and height 30 ft is filled with water to a depth of 15 ft. How much work is required to pump all the water out through a hole in the top of the container if the unit weight of the water is 62.4 lb/ft^3?
Milca Reply
hi am new here I really wants to know how the solve calculus
IBRAHIM
me too. I want to know calculation involved in calculus.
Katiba
evaluate triple integral xyz dx dy dz where the domain v is bounded by the plane x+y+z=a and the co-ordinate planes
BAGAM Reply
So how can this question be solved
Eddy
i m not sure but it could be xyz/2
Leo
someone should explain with a photo shot of the working pls
Adegoke
I think we should sort it out.
Eunice
Eunice Toe you can try it if you have the idea
Adegoke
how
Eunice
a^6÷8
Muzamil
i think a^6 ÷ 8
Muzamil
maxima and minima problem in log form
Roshan Reply
find Maxima and minima of 4^x - 8xlog2
Roshan
(x-1)4^(x-1) - 8log2 4^(x - 1) x - 4^(x - 1) - 8 log(2) -1 to 3 or -11 to 13.
James
thanks
Roshan
The f'(4)for f(x) =4^x
Alice Reply
Practice Key Terms 7

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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