Exponential Logarithmic Exponential Functions fx ax Domain Three

Exponential/ Logarithmic

Exponential Functions f(x) = ax Domain (-∞, ∞) Three types: 1) if 0 < a < 1 2) if a = 1 3) if a > 1 Range (0, ∞)

Laws of Exponents a x + y = a x ay ax/ ay = a x –y (ax)y = axy (ab)x = axbx

Sketching Example Sketch the function y = 3 – 2 x

Exponential Functions are One to One Has an inverse f-1 which is called the logarithmic function (loga) f-1(x) = y f(y) = x ay = x logax = y

Example Find: log 10(0. 001) log 216

Log Graph Reflection of exponential function about the line y = x Domain (0, ∞) Range (-∞, ∞)

Laws of Logarithms loga(xy) = logax + logay loga(x/y) = logax – logay logaxr = rlogax

Example Evaluate log 280 – log 25

e y = ax Many formulas in calculus are greatly simplified if we use a base a such that the slope of the tangent line at y = 1 is exactly 1 For y = 2 x, slope at y = 1 is. 7 For y = 3 x, slope at y = 1 is 1. 1 Value of a lies between 2 and 3 and is denoted by the letter e e = 2. 71828

Example Graph y = ½ e-x – 1 and find the domain and range

Natural log (ln) Log with a base of e logex = lnx = y ey = x

Properties of Natural Logs ln(ex) = x elnx = x ln e = 1

Example Find x if lnx = 5

Example n Solve e 5 – 3 x = 10

Example n Express ln a + ½ ln b as a single logarithm

Expression y = logax ay = x ln ay = ln x y ln a = ln x y = ln x/ ln a logax = ln x/ ln a if a ≠ 0

Example n Evaluate log 85

Example n The half-life of a radioactive substance given by f(t) = 24 ∙ 2 -t/25 Find the inverse