1. 15.2 Logarithmic Functions
OBJ:  To find the base-b logarithm of a positive number
 To find a base-b logarithm equation

2. Graph y = 2 x - 3 – 1 Domain: Range:  (-1, ) 2. Hor/Vert 3 R, 1
 (-1, )
2. Hor/Vert
3 R, 1
3. Inc/Dec
Inc
Key Point ( , )
(3, 0)
Equation of asymp.
y = -1

3. Graph y = 2 x - 3 – 1 Domain: Range:  (-1, ) 2. Hor/Vert 3 R, 1
 (-1, )
2. Hor/Vert
3 R, 1
3. Inc/Dec
Inc
Key Point ( , )
(3, 0)
Equation of asymp.
y = -1

4. Graph y = log 2 ( x + 3 ) – 2 Domain: Range: (-3, )  2. Hor/Vert
(-3, ) 
2. Hor/Vert
3 L 2 
3. Inc/Dec
Inc
Key Point ( , )
(-2, -2)
5. Equation of asymp.
x = -3

5. Graph y = log 2 ( x + 3 ) – 2 Domain: Range: (-2, )  2. Hor/Vert
(-2, ) 
2. Hor/Vert
3 L 2 
3. Inc/Dec
Inc
Key Point ( , )
(-2, -2)
5. Equation of asymp.
x = -3

6. Solve. HW 3 (35-38) 9 = r – 2/3 32 = r – 2/3 (32)-3/2 = (r – 2/3)-3/2
z 5/2 = 243
z 5/2 = 3 5
(z5/2)2/5= (3 5)2/5
z = 32
z = 9

7. DEF:  Base – b logarithm If y = b x , then log b y = x If answer = b exponent , then log base answer = exponent
EX:  log 3 9
log 3 9 = x
3 x = 9
3 x = 3 2
x = 2
 log 3 81
log 3 81 = x
3 x = 81
3 x = 3 4
x = 4
 log 3 1
log 3 1 = x
3 x = 1
3 x = 3 0
x = 0
 log 3 1/3
log 3 1/3 = x
3 x = 1/3
3 x = 3 -1
x = - 1

8. log 3 3 log 3 3 = x 3 x =  3 3 x = 3 ½ x = ½ log 3 1/27
EX: 1  log
P 392 log = x
10 x = 100
10 x = 102
x = 2
 log 21/16
log 21/16 = x
2 x = 1/16
2 x = 2- 4
x = - 4
EX 2  log 3 4  3
P log 3 4 3 = x
3 x = 4 3
3 x = 3 ¼
x = 1/4
 log 2 5 8
log 2 5 8 = x
2x = (23)1/5
x = 3/5

9. Find each logarithm. Solve each equation.
EX:  log 3 5 81
log 3 5 81 = x
3x = 5 81
3x = 34/5
x = 4/5
log 1/3 1/9
log 1/3 1/9 = x
(1/3)x = 1/9
(1/3)x = (1/3)2
x = 2
log 1/3 9
log 1/3 9 = x
(1/3)x = 9
3-x = 32
x = -2
EX:  log b 10,000 = 4
b4 = 10,000
b4 = 104
b = 10
 log 3 y = 5
35 = y
243 = y
EX:  1/81 = b– 4
3– 4 = b– 4
3 = b
1/8 = b – 3
(2)– 3= b– 3
2 = b

10. Solve each logarithmic equation for the variable.HW4 P393 (1-28 all)
EX: 3  log b 81 = 4
P b 4 = 81
b 4 = 3 4
b = 3
log = x
5 x = 625
5 x = 5 4
x = 4
log 4 y = 3
4 3 = y
y = 64
log b1/8 = - 3
b -3 = 1/8
b -3 = 2 -3
b = 2

11. Find each logarithm. EX:  log 5 625 log 5 625 = x 5 x = 625 5 x = 5 4

12. P 394  If y = 2 x what is y – 1 (y inverse)?
x = 2 y
log2x = log 22 y
log2x = y
y = log2x
DEF:  Inverse of an exponential function
y = logbx
NOTE: Since y = 3 x and y = log 3 x are
inverse functions, they are
symmetric with respect to y = x
and the graph of y = 3 x could be
reflected in y = x to obtain
y = log 3 x

13. Rewrite the logarithmic equations in exponential form and the reverse procedure.
EX:  a = log b c; t = b v
b a = c; v = log b t
4 = log 2 16
2 4 = 16
9/16 = (3/4)2
2 = log ¾ 9/16
 5 = log b 70
b5 = 70
 60 = 4 q
q = log 4 60