I.EXPONENTIAL FUNCTION
A. Definition
An exponential function is a function defined by f(x) = ax , where a > 0 and a ≠ 1. The domain of the function is the set of real numbers and the range is the set of positive numbers.
B. Evaluating Exponential Functions
1. Given: f(x) = 2x, find
a. f(3) = ____b. f(5) = _____c. f(2) = ______ d. f(4) = ______ 2. Evaluate f(x) = ( 1)x if
2
a. x = 2 ____b. x = 4 _____c. x = 3 ______ d. x = 4 _______
C. Graphing Exponential Functions
On the same Cartesian Coordinate plane, sketch the graphs of each set of exponential function 1. a) f(x) = 2xb. f(x) = 3xc. f(x) = 4x
2. a) f(x) = 2xb. f(x) = 2x + 1c. f(x) = 2x – 1
3. a) f(x) = 2xb. f(x) = 2x + 2c. f(x) = 2x – 3
4. a) g(x) = 2 xb. g(x) = 3 xc. g(x) = 4 – x
D. The Property of Equality for Exponential Equations
Let a, b and c be real numbers and a≠ 0, then ab = ac if and only if b = c Examples:
1. 32x = 362. 23x = 83. 643x = 84. 10 x = 1/10000
5. 43x = 16x + 26. 16 x = 1/647. 93x = [ 1/3]58.5x+2 – 5x + 1 + 5x = 2625
Do as directed:
A. Evaluate:
1. If f(x) = 3x, what is
a. f( 3)?= ____ b. f( 4) = _____c. f( 2) = _____ d. f( 4) = _____ 2. What is g(x) = [ 1/3 ]x if
a.x = 2 ____ b. x = 4 ______c. x = 3 ______ d. x = 4 ______
B. Solve for x
3. 2x = 1286. 243x = 3
4. 3x = 817. 2x + 2 + 2x + 1 + 2x = 896
5. 42x = 8x + 1 8. 272x – 2 = 95 – x
C. Challenge!!!
9. If x is real and x64 = 64, what is x32?
10. Find the value of xy if 2x = 7 and 7y = 64.
11. If 183 = 2x •3y, find the integer values of x and y. 12. There are about 1,000 bacteria in a certain culture. If the amount doubles every 2 Hours, about how many bacteria would there be after 8 hours?
13. A radioactive substance is decaying (it is changing into...
...A function is a relation in which each element of the domain is paired with exactly one element in the range. Two types of functions are the exponentialfunctions and the logarithmicfunctions. Exponentialfunctions are the functions in the form of y = ax, where ''a'' is a positive real number, greater than zero and not equal to one. Logarithmicfunctions are the inverse of exponentialfunctions, y = loga x, where ''a'' is greater to zero and not equal to one. These functions have certain differences as well as similarities between them. Also they are very useful for various situations in life.
Logarithmicfunctions are fairly different from the exponentialfunctions. The first difference that we can find between them is in the equations, they are inverse to each other. The logarithmic equation is y = loga x and the exponential equation is y = ax. We can also see that the natural exponentialfunction is different form the natural logarithmicfunction. The natural exponentialfunction is y = f(x) = ex and the natural logarithmicfunction is f(x) = loge x = lnx , where x > 0. Also we can...
...Exponential and LogarithmicFunctions
* Verify that the natural logarithm function defined as an integral has the same properties as the natural logarithm function earlier defined as the inverse of the natural exponentialfunction.
Integrals of Exponential and LogarithmicFunctionsFunction  Integral 
lnx  x ∙ lnx  x + c 
logx  (x ∙ lnx  x) / ln(10) + c 
logax  x(logax  logae) + c 
ex  ex+c 
ek∙x  1 / k ∙ ek∙x + c 
ax  ax / lna + c 
xn  1 / (n+1) ∙ xn+1 + c, where n≠ 1 
1/x = x1  lnx+c 
√x = x1/2  2/3 ∙ (√x)3 + c = 2/3 ∙ x3/2 + c, where c is a constant 
Example 1: Solve integral of exponentialfunction ∫ex32x3dx
Solution:
Step 1: the given function is ∫ex^33x2dx
Step 2: Let u = x3 and du = 3x2dx
Step 3: Now we have: ∫ex^33x2dx= ∫eudu
Step 4: According to the properties listed above: ∫exdx = ex+c, therefore ∫eudu = eu + c
Step 5: Since u = x3 we now have ∫eudu = ∫ex3dx = ex^3 + c
So the answer is ex^3 + c
Example 2: Integrate .
Solution: First, split the function into two parts, so that we get:
Trigonometric Identities and Ratio and Proportion
Trigonometric Identities
In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x =x" or usefully...
...LOGARITHMIC AND EXPONENTIALFUNCTIONS
Inverse relations
ExponentialfunctionsExponential and logarithmic equations
One logarithm
THE LOGARITHMICFUNCTION WITH BASE b is the function
y = logb x.
b is normally a number greater than 1 (although it need only be greater than 0 and not equal to 1). The function is defined for all x > 0. Here is its graph for any base b.
Note the following:
• For any base, the xintercept is 1. Why?
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
The logarithm of 1 is 0. y = logb1 = 0.
• The graph passes through the point (b, 1). Why?
The logarithm of the base is 1. logbb = 1.
•  The graph is below the xaxis  the logarithm is negative  for 

 0 < x < 1. 

 Which numbers are those that have negative logarithms? 
Proper fractions.
•  The function is defined only for positive values of x. 

 logb(−4), for example, makes no sense. Since b is always positive, no power of b can produce a negative number. 
• The range of the function is all real numbers.
• The negative yaxis is a vertical asymptote (Topic 18).
Example 1. Translation of axes. Here is the graph of the natural logarithm, y = ln x (Topic 20)....
...CHAPTER 1
1.1 Introduction
Exponential and logarithms functions are important concepts that play crucial roles in college mathematics courses, including calculus, differential equations, and complex analysis. The purpose of this study is to describe a theory of how students might develop their understanding of these topics and to analyze understanding of these concepts within the context of this theory, their application in real life phenomena and discussing with their model.
One of the basic principles studied in mathematics is the observation of relationships between two connected quantities. A function is this connecting relationship, typically expressed in a formula that describes how one element from the domain is related to exactly one element located in the range (Lial & Miller, 1975). The exponentialfunction is not to be confused with the polynomial functions, such as x2. One way to recognize the difference between the two functions is by the name of the function. Exponentialfunctions are called so because the variable lies within the exponent of the function (Allendoerfer, Oakley, & Kerr, 1977). These functions are often recognized by the fact that their rate of growth is proportional to their value (Bogley & Robson, 1999). This concept of exponential growth has been around...
...MATH133 Unit 5: Exponential and LogarithmicFunctions
Individual Project Assignment: Version 2A
Show all of your work details for these calculations. Please review this Web site to see how to
type mathematics using the keyboard symbols.
IMPORTANT: See Question 1 in Problem 2 below for special IP instructions. This is
mandatory.
Problem 1: Photic Zone
Light entering water in a pond, lake, sea, or ocean will be absorbed or scattered by the particles
in the water and its intensity, I, will be attenuated by the depth of the water, x, in feet. Marine
life in these ponds, lakes, seas, and oceans depend on microscopic plant life that exists in the
photic zone. The photic zone is from the surface of the water down to a depth in that particular
body of water where only 1% of the surface light remains unabsorbed or not scattered. The
equation that models this light intensity is the following:
𝐼 = 𝐼0 𝑒 −𝑘𝑥
In this exponentialfunction, I0 is the intensity of the light at the surface of the water, k is a
constant based on the absorbing or scattering materials in that body of water and is usually called
the coefficient of extinction, e is the natural number 𝑒 ≅ 2.718282, and I is the light intensity at
x feet below the surface of the water.
1. Choose a value of k between 0.025 and 0.095.
2. In a lake, the value of k has been determined to be the value that you chose above, which
means that 100k% of the surface...
...Solving Exponential and Logarithmic Equations
Exponential Equations (variable in exponent position)
1. Isolate the exponential portion ( base exp onent ): Move all nonexponential factors or terms to the other side of the equation. 2. Take ln or log of each side of the equation. • Make sure to use ln if the base is “e”. Then remember that ln e = 1 . • Make sure to use log if the base is 10. • If the base is neither “e” nor “10”, use either ln or log, your choice.. 3. Bring the power (exponent) down into coefficient position. 4. Use various algebra techniques to solve for the variable. 5. Check your answer by evaluating the original equation with your calculator.
Example: 4e 2 x −3 = 40
Answer is on next page
Logarithmic Equations
1. 2. 3. 4. 5. Move all log terms to one side of the equation, all nonlog terms to the other side. Combine log terms into a single log term using the laws of logarithms. Write the log equation in its exponential form. (remember: 2 3 = 8 ↔ log 2 8 = 3 ) Use various algebra techniques to solve for the variable. Check your answer using your calculator. Remember that domain problems occur in log functions. • If the base of the log is “10” or “e”, you can use the appropriate calculator keys. • If the log is not “10” or “e”, you may need to use the change of base formula before using your calculator.
Example:
log 5 ( x + 1)...
...CHAPTER 2
EXPONENTIAL AND LOGARITHMS FUNCTIONS (534)
2.1 Exponentialfunctions and their graph
Definition
An exponentialfunction with base b is defined by the equation
[pic] Or [pic] ([pic] [pic]and x is a real number)
The domain of any exponentialfunction is the interval ([pic] The range is the interval [pic]
Example 1: Graph theexponentialfunction [pic]
Example 2: Graph the exponentialfunction [pic]
Properties of exponentialfunctions
1. The domain of any exponentialfunction is the interval ([pic]
2. The range is the interval [pic]
3. The [pic]intercept is (0,1).
4. The [pic]axis is an asymptote of the graph.
5. The graph of [pic]passes through the point [pic]
2.2 Logarithms functions and their graphs
Example 1: Graph the logarithms function [pic]
x f(x) 
0.25 2 
0.5 1 
1 0 
2 1 
4 2 
Example 2: Graph the logarithms function [pic]
x f(x) 
0.5 2 
0.5 1...
...Alg2  CH7 Practice Test
Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Tell whether the function y = 2( 5 ) shows growth or decay. Then graph the function. a. This is an exponential growth function. c. This is an exponential decay function.
x
b. This is an exponential growth function. d. This is anexponential growth function.
____
2. Graph the inverse of the relation. Identify the domain and range of the inverse. x y
−1
4
1 2
3 1
5 0
7 1
a.
c.
Domain: {x  0 ≤ x ≤ 4}; Range: {y  − 1 ≤ y ≤ 7} b. d.
Domain: {x  − 1 ≤ x ≤ 7}; Range: {y  − 4 ≤ y ≤ 1}
____
Domain: {x  − 7 ≤ x ≤ 1}; Domain: {x  − 7 ≤ x ≤ 1}; Range: {y  0 ≤ y ≤ 4} Range: {y  − 4 ≤ y ≤ 0} 3. Manny is a plumber and charges $50 when he visits a client and $30 per hour for every hour he works. His bill can be expressed as a function of hours, x, with the function f(x) = 50 + 30x. Which statement explains the meaning of the inverse of the function? a. Total bill as a function of the number of hours b. Cost per hour as a function of the total bill c. Number of hours as a function of the total bill d. Total bill as a function of the cost per hour
3 4. Write the exponential equation 2 = 8 in...