Exponential+Logarithmic+Functions

=Exponential & Logarithmic Functions= By: Daniel Hidalgo, Ricardo Flor, Bernardo Missura


 * 1) Exponential Functions
 * Transformations
 * Basic Shapes
 * Standard Equation
 * Domain, Range
 * Asymptotes
 * Detailed variable changes and effects
 * Graphs on each situation
 * 1) Logarithmic Functions
 * Transformations
 * Basic Shapes
 * Standard Equation
 * Domain, Range
 * Asymptotes
 * Detailed variable changes and effects
 * Graphs on each situation

Exponential and Logarithmic functions are opposites and transformations. The different variables the basic equations have are 5 that are set to be (a, b, c, h and k) These variables manipulate different aspects of the graph and set the intercepts, asymptotes, domain, range, steepness, and other aspects that will be seen more profoundly later on.

Basic Equation:
 * Exponential Functions:**


 * //y = a*c^b(x-h) + k//**

With: f(x) = **__1*2^(0*(x - 0)) + 0__**

* (h): Changing the variable h in the equation for exponential functions manipulates the point where the equation will change. It changes along the x-axis only. Negative numbers will start the curve in the negative x integers, and positive numbers will start the curve in the positive x-axis side. Negative numbers go along the left and positive numbers to the right of the x-axis


 * Domain: All real numbers


 * H positive Value** **H negative Value**



*(k): K determines the movement along the y-axis, and therefore the asymptotes. Just like h makes up the movements along the x-axis to the right or left, the k variable determines the movement of t the function along the y-axis. If k is 1, then the asymptote will be at y=1 because there will be where it ends.(y-intercept)


 * Domain: all real numbers


 * K Positive Value K Negative Value**

*(a): Changing the coefficient a in the slider changes the curve’s y intercept. For example if a is 1, then when x=0, the y intercept would be 1 and from then on it will continue increasing to the exponent the function is elevated to. Positive values for a have the function’s shape on the positive y-axis while negative a values have all negative y-axis values.

* Domain: All Real numbers * At x=0 the line is straight


 * A Positive Value A Equals 0 Value A Negative Value **

*(b): Changing the exponent (b) using the slider changes the direction of the function along the x-axis. It is also what manipulates the steepness of the slope itself in terms of x to the power it is in. When b has positive integers, the higher the integer, the closest the curve will be to the y-axis. The closer to 0 the value is, the farthest away the curve will be from the y-axis. The same thing happens with negative values but in this case the closer to 0 the integer is, the farther away from the y-axis, and the farther away the integer from 0 the closer to the y-axis. When b=0, the whole function is a horizontal line in y=1. As explained before, it changes the steepness of the curve.


 * Domain: X does not equal 1 or -1
 * At X=0 theres a straight line


 * b Positive Value b Equals 0 b Negative Value**

*(c): Changing the c values gives the curve the place where it will be located. Negative b value will place the curve in the negative y axis while positive b will locate the curve on the positive y axis. B=0 will palce a straight line in y=0 line with a range on only positive x range. In this case, the a value will give the value in which the curve will cross the y intercept as it increases exponentially.


 * Domain: All Real Numbers
 * Range: X greater than or equal to 0


 * c Positve Value c Equals 0 c Negative Value**

With f(x) = **__2*(1/2)^x__.**

First of all, (a) tells us what the y intercept is in the graph. Also, changing it varies how long is the function and it stretches it. At point 0, the function is a straight line right along the x-axis. When it starts increasing closer to 1, it has a shape and (a) manipulates how stretched or collected is the curve. It is like unfolding or collecting a mat in terms of displacement along the x-axis.

With f(x) = **__1*2^(1*(x - 0)) + 0__**

There are changes in the function in terms of steepness again. Variable (b) manipulates the precipitation the function has and does not change the y intercept while the value (a) analyzed before. The difference between these two is that one has to do with the displacement along the axis (whichever axis) and the other ones have to do with the actual shape (does not change the displacement of the curve but the steepness).

General Transformations:


 * h - Translate Graph Horizontally
 * b - Horizontal Stretching or Compression
 * a - Vertical Stretching or Compression
 * k - Translate Graph Vertically (y-intercept)

Scientists use exponents to measure the size of earthquakes, also in the computer system you often see gigabytes, megabytes, nano-bytes, so mega means 10^ 6 and giga means 10^ 9. Another example is to find a certain number, after seeing variables, for example you want to see how many soccer balls 6 people can make in 6 days, when 2 people can make 6 balls in 1.5 days. You can also use exponent functions, to solve compound interests examples such as population growths because it depends on the growth proportion in different years (it will grow by the same percentage, but in essence the quantity increased will be more each year).
 * 1) Real Life examples of exponents:


 * Logarithmical Functions:**
 * Basic Equation:**


 * //y = (a*c^h/b)// + k**

1 is a, log(1) is h, x is variable, (-0) is h, log(2) is b and 0 is k
 * With:**
 * f(x) = __(1*log(1*(x-0))/log(2)+0 __**

Logarithmic Functions

Basic Equation:

"log(base a) x=n meaning a^(n)=x" this basic equation is really easy, because it is only another way to write an exponent, for example: 10^(4)= 10 000; is the same as log(base 10) 10 000=4

Analyze the graph, with this equation: f(x) = 1*log(1*(x-0))/log(2)+0 1*log(1*(x-0))/log(2)+0, and in this graph the variables are: h*log(k*(x-b))/log(a)+c.

k, b, a and c are the variables, where we change the numbers and see how those changes affect the graph.


 * "h": Changing the values for h will affect the graph in the palce where the curve will cross the x axis. The higher the value, the more to the positive x axis the x intercept will be. Positive and 0 value for h will have the x-intercept in the positive x axis while all negative values except -1 will have the x intercept in the negative x range. h= -1 will ahve the x-intercept at y=0 and x=0.
 * Domain: All Real Numbers
 * h value will give the lines assymptote so if h=-2, the line will go from x greater -2 and on
 * h value will give the lines assymptote so if h=-2, the line will go from x greater -2 and on


 * h Positive Value h Negative Value**




 * "k": .The change in this variable in the equationa ffects the graph by changing the y intercept. Depending on the value of k, the line will start at a different point in the y axis. It is important to see that there is an assymptote in the x=0 place. The palce value where the line will start is determined by substracting 3 numbers from the value of k.


 * Domain: All Real Numbers
 * Assymptote on x=0


 * k Positive Value k Negative Value**


 * "b": the change in b will also give the steepness of the curve. But in this case the values must be greater than 1 because there is no log of a negative number. The curve will always have its curve towards the positive x axis and the greater the value of b, the smaller the slope of the curve. At x=0 there will be a straightline.


 * Domain: X does not equal to -1 or 1.
 * X=0 there is a straight line


 * [[image:b_pos.png]]e**[[image:b=0.png width="316" height="419"]]


 * "a": The change in the value of a will influence the steepness and direction of the curve. The negative values plugged in as a will cause the line to increase negatively towards negative y axis having its assymptote at x=0 on the positive y range. If the value is positive, the curve will grow towards the positive y range and positive x range while it will have its assymptote towards the negative y axis in x=0. The greater the number, the steeper the slope of the graph.


 * Domain: X greater than 1
 * X=0 is straight line
 * assymptote at x=0


 * a Equals 0**


 * "c": The value of c gives first of all the direction of the lines curve. If the values are negative, the line will icnrease to the neagtive x axis, if the value is positive the line will increase towards the positive x axis. Besides, as the value of c increases, the line increases the y value at which the assymptote starts. The starting value of y of the assymptote x=0 is greater as c value increases magnitud be it in negative or positive direction.


 * Domain: X greater than 0
 * x greater than 0 as its inverse function has y greater than 0 results values
 * X cannot equal 0


 * c Value Positive c Value Negative**

General Transformations:
 * h: It transforms the steepness and the stretching of the line.
 * k: It translates the graph vertical, and horizontal depending on the value.
 * a: it translates the line horizontally.
 * b: When the value is bigger than one it compresses it or stretch it.
 * c: It translates vertically.

In conclusion, no matter which function we use we can see that the transformations work all the same way for any type of function as seen with these two function examples.

A good example of using logarithmic function in real life examples, is by an aerospace engineer, if he wants to graph the lift of the wing vs. the size of the wing, he can use many ways and methods, in order to find what he wants to. Also measuring the earthquakes scale, "Richter scale". Also the loudness of sound
 * 1) Real Life Applications:

2. Another real life example of the application of logarithms is the pH values of liquids. Based on 10, the values of Potential hydroniums increase or decrease in 1 value for every 10^x incresae or decrease.

Bibliography:


 * Exponential Functions: http://oregonstate.edu/instruct/mth251/cq/FieldGuide/exponential/lesson.html
 * Logarithmic Functions: http://math.asu.edu/fym/Courses/mat117_web/exponential_and_logarithmic_functions_notes/laws-of-logarithms/Laws_of_Logarithmic_Functions.html
 * Transformations (Visual Calculus): http://archives.math.utk.edu/visual.calculus/0/shifting.5/index.html
 * Miller, Maria. "Where Do You Need Exponents in Everyday Life?" // Homeschool Math - Free Math Worksheets, Ebooks, Lessons, Curriculum Guide. // Web. 26 Mar. 2011. .
 * ""Real-World" Application for Specific Mathematical Topics." // Math Central //. Web. 26 Mar. 2011. .

Exponential Logarithmic Functions In the real world examples.

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