Logarithms

Logarithmic Functions x = 2y 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 x = ay Logarithmic Equation y = loga x 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 loga64 = 2 base number exponent a2 = 64 a = 8 Example: Solve log5 x = 3 Rewrite in exponential form: 53 = x x = 125

Example: Solve 7y = 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: a. log8 84 b. 6[log6 (3y – 1)] logb bx = x log8 84 = 4 blogb x = x 6[log6 (3y – 1)] = 3y – 1 Here are some special logarithm values: 1. loga 1 = 0 because a0 = 1 2. loga a = 1 because a1 = 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) = log3 x This is the inverse of g(x) = 3x g(x) = 3x x g(x) x f(x) -2 -1 1 2 1/9 1/3 1 3 9 1/9 1/3 1 3 9 -2 -1 1 2 f(x) = log3 x

More Logarithmic Functions A logarithmic function is the inverse of an exponential function. For the function y = 2x, the inverse is x = 2y. In order to solve this inverse equation for y, we write it in logarithmic form. x = 2y is written as y = log2x and is read as “y = the logarithm of x to base 2”. y = 2x 1 2 4 8 16 1 2 4 8 16 y = log2x (x = 2y)

Graphing the Logarithmic Function y = x y = 2x y = log2x

Comparing Exponential and Logarithmic Function Graphs y = 2x y = log2x The y-intercept is 1. There is no y-intercept. There is no x-intercept. The x-intercept is 1. The domain is {x | x Î R}. The domain is {x | x > 0}. The range is {y | y Î R}. The range is {y | y > 0}. There is a horizontal asymptote at y = 0. There is a vertical asymptote at x = 0. The graph of y = 2x has been reflected in the line of y = x, to give the graph of y = log2x. This is because logarithmic functions are inverses of exponential functions

2 is the exponent of the power, to which 7 is raised, to equal 49. 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 (log749 = 2). Exponential notation Logarithmic form log749 = 2 72 = 49 In general: If bx = N, then logbN = x. State in logarithmic form: State in exponential form: a) 63 = 216 log6216 = 3 a) log5125 = 3 53 = 125 b) 42 = 16 log416 = 2 b) log2128= 7 27 = 128

Evaluating Logarithms Note: log2128 = log227 = 7 log327 = log333 = 3 log2128 = x 2x = 128 2x = 27 x = 7 log327 = x 3x = 27 3x = 33 x = 3 3. log556 = 6 logaam = m 4. log816 5. log81 log816 = x 8x = 16 23x = 24 3x = 4 log81 = x 8x = 1 8x = 80 x = 0 loga1 = 0

Evaluating Logarithms 6. log4(log338) 7. = x log48 = x 4x = 8 22x = 23 2x = 3 2x = 1 9. Given log165 = x, and log84 = y, express log220 in terms of x and y. 8. log165 = x log84 = y = 23 = 8 16x = 5 24x = 5 8y = 4 23y = 4 log220 = log2(4 x 5) = log2(23y x 24x) = log2(23y + 4x) = 3y + 4x

Evaluating Base 10 Logs Base 10 logarithms are called common logs. Using your calculator, evaluate to 3 decimal places: a) log1025 b) log100.32 c) log102 1.398 -0.495 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.