Logarithmic Functions y x 2 is an exponential

Logarithmic Functions y x = 2 is an exponential equation. If we solve for y it is called a logarithmic equation. Let’s look at the parts of each type of equation: Exponential Equation Logarithmic Equation y y = loga x x=a exponent /logarithm base number In General, a logarithm is the exponent to which the base must be Raised to get the number that you are taking the logarithm of.

Example: Rewrite in exponential form and solve loga 64 = 2 base number exponent 2 a = 64 a= 8 Example: Solve log 5 x = 3 Rewrite in exponential form: 3 5 =x x = 125

Example: Solve y 7 = 1 49 y = – 2 An equation in the form y = logb x where b > 0 and b ≠ 1 is called a logarithmic function. Logarithmic and exponential functions are inverses of each other logb bx = x blogb x = x

Examples. Evaluate each: 4 a. log 8 8 x logb b = x 4 log 8 8 = 4 b. [log 6 (3 y – 1)] 6 logb x b =x [log 6 (3 y – 1)] 6 = 3 y – 1 Here are some special logarithm values: 1. loga 1 = 0 because a 0 = 1 2. loga a = 1 because a 1 = a 3. loga ax = x because ax = ax

How do you graph a logarithmic function? We will need to create a table of values. (Keep in mind that logarithmic functions are inverses of exponential functions) Example: Graph f(x) = log 3 x This is the inverse of g(x) = 3 x g(x) x f(x) -2 -1 0 1 2 1/9 1/3 1 3 9 -2 -1 0 1 2 x g(x) = 3 x f(x) = log 3 x

A logarithmic function is the inverse of an exponential function. For the function y = 2 x, the inverse is x = 2 y. In order to solve this inverse equation for y, we write it in logarithmic form. x = 2 y is written as y = log 2 x and is read as “y = the logarithm of x to base 2”. y = 2 x y = log 2 x (x = 2 y) 1 2 4 8 16

Graphing the Logarithmic Function y=x y = 2 x y = log 2 x

Comparing Exponential and Logarithmic Function Graphs y = 2 x The y-intercept is 1. There is no x-intercept. The domain is {x | x Î R}. y = log 2 x The range is {y | y > 0}. There is no y-intercept. The x-intercept is 1. The domain is {x | x > 0}. The range is {y | y Î R}. There is a horizontal asymptote at y = 0. There is a vertical asymptote at x = 0. The graph of y = 2 x has been reflected in the line of y = x, to give the graph of y = log 2 x. This is because logarithmic functions are inverses of exponential functions

Logarithms Consider 72 = 49. 2 is the exponent of the power, to which 7 is raised, to equal 49. The logarithm of 49 to the base 7 is equal to 2 (log 749 = 2). Exponential notation Logarithmic form 72 = 49 log 749 = 2 In general: If bx = N, then logb. N = x. State in logarithmic form: State in exponential form: a) 63 = 216 log 6216 = 3 a) log 5125 = 3 53 = 125 b) 42 = 16 log 416 = 2 b) log 2128= 7 27 = 128

Evaluating Logarithms 1. log 2128 = x 2 x = 128 2 x = 27 x=7 3. log 556 = 6 4. log 816 = x 8 x = 16 23 x = 24 3 x = 4 2. log 327 = x 3 x = 27 3 x = 33 x=3 Note: log 2128 = log 227 =7 log 327 = log 333 =3 logaam = m 5. log 81 = x 8 x = 1 8 x = 80 x=0 loga 1 = 0

Evaluating Logarithms 6. log 4(log 338) log 48 = x 4 x = 8 22 x = 23 2 x = 3 8. = 23 =8 7. =x 2 x = 1 9. Given log 165 = x, and log 84 = y, express log 220 in terms of x and y. log 165 = x log 84 = y 16 x = 5 8 y = 4 24 x = 5 23 y = 4 log 220 = log 2(4 x 5) = log 2(23 y x 24 x) = log 2(23 y + 4 x) = 3 y + 4 x

Evaluating Base 10 Logs Base 10 logarithms are called common logs. Using your calculator, evaluate to 3 decimal places: a) log 1025 1. 398 b) log 100. 32 -0. 495 c) log 102 0. 301

Your Task To receive credit for introduction to logs you must complete a written assignment. Your teacher will give you the assignment from a resource your school recommends.