1. 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 = ) 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

2. 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.

3. 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.

4. 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) y = - ½ log4 (x + 3) - 7
V stretch V shrink and flip over x-axis
Crit pt: (2 + 1 , 4) = (3,4)
Asy: x = Crit pt: (–3 + 1 , –7) = (–2,–7)
Asy: x = –3