e/ Natural Log

e y = a x 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 =

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

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

Properties of Natural Logs ln(e x ) = x e lnx = x ln e = 1

Example Find x if lnx = 5

Example Solve e 5 – 3x = 10

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

Expression y = log a x a y = x ln a y = ln x y ln a = ln x y = ln x/ ln a log a x = ln x/ ln a if a ≠ 0

Example Evaluate log 8 5

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