Logarithmic Functions and Their Graphs I.. Logarithm = another way of writing exponential expressions. A) log base# x = y B) Converting log expressions into exponent form (vice-versa). 1) log b y = x bx = y Examples: 1) Log 4 64 = x  4x = 64 2) 34 = x  log 3 x = 4 C) Natural log = a log with a base of e. (symbol = ln) D) Basic shape of y = logbx critical pt at (1,0) Where 0< b<1 Crit (1,0) Crit (1,0) Where b > 1

Logarithmic Functions and Their Graphs II.. Log & Exponential functions are Inverses of each other. A) log b x = y is the same as by = x 1) And the inverse of by = x is bx = y … 2) But bx = y is a normal growth or decay graph 3) Then log b x = y is the inverse of bx = y. a) Find (x,y) values for bx = y and inverse them to get log b x = y B) Therefore, logs are the inverse of an exponential function. 1) It reflects about the line y = x. 2) The original (x,y) coordinates become (y,x) for inverse. 3) The horizontal asy. y = # becomes a vertical asy. x = #. C) To graph log functions, you can graph the inverse of its expo function OR just shift the logbase x = y parent function.

Logarithmic Functions and Their Graphs III. Sketching Logarithm Graphs y = A• logbase (x +C) + D A) y = logb x + D 1) Moves the graph up/down. + is up, – is down. B) y = logb (x + C) 1) Moves the graph left/right. + goes , – goes 2) Also moves the vertical asymptote. Asy: x = -C C) y = A • logb x (the “A” term is the “slope”) 1) If A > 1, then the graph gets taller (Vert stretch). 2) If 0 < A < 1, then the graph gets shorter (Vert shrink). 3) If A is negative, the graph will also flip over the x-axis. D) Shifted Critical Point = (–C + 1, D). Use magic paper for flips.

Logarithmic Functions and Their Graphs Examples of shift rules y = A• logbase (x + C) + D Shifted Critical Point = (–C + 1, D). y = 5 log8 (x – 2) + 4 y = - ½ log4 (x + 3) - 7 V stretch 2 4 V shrink and flip over x-axis Crit pt: (2 + 1 , 4) = (3,4) 3 7 Asy: x = 2 Crit pt: (–3 + 1 , –7) = (–2,–7) Asy: x = –3