Logarithmic Functions Lesson 3-2 Logarithmic Functions And Their Graphs

Logarithmic Functions and Their Graphs When one looks at the graph of the exponential function y = ax, by applying the horizontal line test, one can see that this has an inverse function. This inverse function is called a logarithm. The sole function of a logarithm is to find an exponent! Read the history of logarithms found at: http://www.spiritus-temporis.com/logarithm/history.html

Logarithmic Functions and Their Graphs Definition of Logarithmic Function For x > 0, a > 0, and a  1, y = loga x if and only if x = a y. The function given by f(x) = loga x is called the logarithmic function with base a.

Logarithmic Functions and Their Graphs Examples: 1. Evaluate f(x) = log4x when x = 64. Solution: f(64) = log464 which means 4 to what power equals 64? 43 = 64 Therefore, f(64) = log464 = 3 (Remember that the purpose of a logarithm is to find an exponent!)

Logarithmic Functions and Their Graphs Example 2: Evaluate f(x) = log5x when x = 1/25. Solution:

Logarithmic Functions and Their Graphs Exampe 3: Evaluate f(x) = log2x when x =1. Solution:

Logarithmic Functions and Their Graphs Try the following: Solutions: 1. f(x) = log6x when x = 216 f(216) = log6216 = 3 2. f(x) = log4x when x = 1 f(1) = log41 = 0 3. f(x) = log4x when x = 1/1024 f(1/1024) = log4(1/1024) = - 5

Logarithmic Functions and Their Graphs Some Basic Properties of Logarithms:

Logarithmic Functions and Their Graphs Now let’s look at the graph of a logarithmic function. Remember that when a graph has an inverse, the x- and y-coordinates change positions. Look at the following tables. Inverse function y = log3x y = 3x x y -3 1/27 -2 1/9 -1 1/3 0 1 1 3 2 9 3 27 Its inverse is x y 1/27 -3 1/9 -2 1/3 -1 1 0 3 1 9 2 27 3

Logarithmic Functions and Their Graphs Graphing y = log3x

Logarithmic Functions and Their Graphs Now you graph y = log2x

Logarithmic Functions and Their Graphs Now let’s compare both graphs: What is the domain for each graph? What is the range for each graph? What is the x-intercept for each graph? What appears to be the vertical asymptote for each graph? Is the function increasing or decreasing? If so, what is the interval?

Logarithmic Functions and Their Graphs What would the function be with the following transformations? 1. The reflection of y = logax? 2. A horizontal shift of 3 units to the left for y = log4x? 3. A vertical shift of 5 units up for y = log6x? 4. The reflection of y = logax with a horizontal shift of c units and a verical shift of d units?

Logarithmic Functions and Their Graphs The Natural Logarithmic Function If y = ex is the natural exponential function, then y = ln x is the natural logarithmic function. For x > 0, y = ln x if and only if x = ey. The function is given by f(x) = logex = ln x is called the natural logarithmic function.

Logarithmic Functions and Their Graphs Graph the natural exponential function, the natural logarithmic function, and the line of symmetry all on one graph.

Logarithmic Functions and Their Graphs Properties of Natural Logarithms 1. ln 1 = 0 Why? Because e 0 = 1 2. ln e = 1 Why? Because e 1 = e Because in exponential form if the two bases are the same, then the exponents must be equal. 3. ln ex = x Why? 4. If ln x = ln y, then x = y Why? Because since the two bases are the same, the two results must be equal.

Logarithmic Functions and Their Graphs What you should know: What a logarithm is 2. Basic properties for logarithms 3. How to graph a logarithmic function 4. Transformations to a logarithmic function 5. The natural logarithmic function and its basic properties