Section+3.3

Section 3.3: __Logarithmic Functions and Their Graphs.__

1. Switch X and Y, x=b^y 2. Solve for Y In order to find y as an exponent, logarithms are used. If b, x > 0, and b does NOT equal 1, then x=b^y (exponential) if and only if y=log(b)x (logarithmic)
 * Review**: find the inverse of y=b^x
 * Definition:**


 * KEY**: If the letter or number is in parentheses it means it is the subscript of the log.

1. log2(8)=x 2^x=8 2^x=2^3 x=3
 * Example of Changing Logarithmic Functions into Exponential Function Form**:

1. logb (1) = 0 2. logb(b)= 1 3. logb(b)^y= y 4. b^logbx=x
 * Properties of Logarithmic Functions;** For x,b > 0 and b does NOT equal 1, y is a real number

The common base of logarithmic functions is 10. The base is not shown it is default; therefore if a log doesn't have a subscript, the subscript is 10. Example: 1. log100=2 The subscript of the log is 10, 10^2= 100. 2. logx=3 10^3=x x=1000
 * The Common Base:**

base e=ln, ln=natural log. Examples: 1. lne^5 5
 * The Natural Base e:**

1. Logarithmic Growth: This is a graph of logarithmic growth. The parent function is y=log(2)x This graph is the inverse of exponential growth as learned in previous sections.
 * Graphing Logarithmic Functions.**

2. Logarithmic Decay: The graph of logarithmic decay is the same as logarithmic growth except it is reflected over the x-axis. The parent function is y=log(1/2)x

1. b>1 shows growth 2. 0<b<1 shows decay 3. x=o is a vertical asymptote 4. D= (0, infinity) R= all real numbers 5. the function is one-to-one and therefore can have an inverse.
 * Properties of Graphs:**

y= alog(b) (x-h) + k
 * Family of Functions: Explains the Transformations of the graph.**

Example: y= -2log(2) (1-x) +4 Transformations: From left to right 1. Graph is reflected over x-axis 2. Veritcal Stretch of 2 3. log(2) makes the graph a logarithmic growth 4. (1-x) must be switched to [-(x-1)], therefore graph is reflected over y-axis 5. Graph is moved right 1 6. Graph is moved up 4 Use parent function of logarithmic growth and make these transformations. To make graph more clear create an x,y table and find a few points.

pg. 308 # 3-60 mult 3 pg. 308 #1-58 every 3rd. (1,4,7,10 etc...)
 * Homework from Section 3.3**