EXPONENTIAL FUNCTION where (base) b > 0 and b For 0 < b < 1, The graph is increasing and asymptotic to (-) x-axis. The graph is decreasing and asymptotic to (+) x-axis. (0 , 1) is always a point on the graph (0 , 1) (0 , 1) 1

x -2 -1 1 2 f(x) 4 x -2 -1 1 2 f(x) 4 (0 , 1) (-2 , 4) (2 , 4) 1 2 f(x) 4 x -2 -1 1 2 f(x) 4 (-2 , 4) (2 , 4) (-1 , 2) (1 , 2) (0 , 1) (0 , 1) 2

x -2 -1 1 2 f(x) x -2 -1 1 2 f(x) 4 (0 , -2) (-2 , 2) (-2 , 4) 1 2 f(x) x -2 -1 1 2 f(x) 4 (-2 , 2) (-2 , 4) (-1 , 0) (-1 , 2) (0 , -2) (0 , 1) 3

x 1 2 3 4 f(x) x -2 -1 1 2 f(x) 4 (-2 , 4) (-1 , 2) (0 , 1) (0 , 4) 1 2 3 4 f(x) x -2 -1 1 2 f(x) 4 (-2 , 4) (0 , 4) (1 , 2) (-1 , 2) (2 , 1) (0 , 1) 4

x -2 -1 1 2 f(x) 4 x -2 -1 1 2 f(x) 3 4 6 (0 , 1) (2 , 4) (1 , 2) 1 2 f(x) 4 x -2 -1 1 2 f(x) 3 4 6 (2 , 6) (2 , 4) (1 , 4) (0 , 3) (1 , 2) (0 , 1) 5

x -2 -1 1 2 f(x) 4 x -4 -3 -2 -1 f(x) 1 2 4 (0 , 1) (2 , 4) (1 , 2) 1 2 f(x) 4 x -4 -3 -2 -1 f(x) 1 2 4 (2 , 4) (0 , 4) (1 , 2) (-1 , 2) (0 , 1) (-2, 1) 6

x -1 1 f (x) e x -1 1 f (x) e (-1 , e) (1 , e) (0 , 1) (0 , 1) 7

Rules of Exponents

Logarithm of a number N to a positive base b ( ) is the exponent to which b must be raised in order to get N. S.Y.Tan

Properties of Logarithm or Laws of Logarithm S.Y.Tan

LOGARITHMIC FUNCTION for (base) b > 0 and b logarithmic form exponential form Change logarithmic form to exponential form b = e (natural logarithm) b =10 (common logarithm) 11

LOGARITHMIC FUNCTION for (base) b > 0 and b logarithmic form exponential form Change logarithmic form to exponential form 12

LOGARITHMIC FUNCTION for (base) b > 0 and b logarithmic form exponential form Change exponential form to logarithmic form 13

LOGARITHMIC FUNCTION for (base) b > 0 and b logarithmic form exponential form Change exponential form to logarithmic form 14

Use the properties of logarithm to expand each expression. The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. Use the properties of logarithm to expand each expression. 15

Use the properties of logarithm to condense the expression as single logarithm. 16

Evaluate the following. 17

Logarithmic function and Exponential function are inverse functions of one another. 18

Logarithmic function and Exponential function are inverse functions of one another. 19

The graph is decreasing and asymptotic to (+) y-axis. For 0 < b < 1, For b > 1, The graph is decreasing and asymptotic to (+) y-axis. The graph is increasing and asymptotic to (-) y-axis. (1 , 0) is always a point on the graph (1, 0) (1 , 0) 20

(2, 4) (0, 1) (4, 2) (1 , 0) 21

(-1, 2) (0, 1) (1 , 0) (2, -1) 22

(1, e) (0, 1) (e, 1) (1 , 0) 23

Solve for x in terms of y 24

x = -1 y = x y = -1

Solve for x in terms of y 26

y = x y = 2 x = 2

Solve for x in terms of y 28

x = 0 y = 0

Exponential and Logarithmic Equations 30

Exponential and Logarithmic Equations 31

Exponential and Logarithmic Equations 32

Exponential and Logarithmic Equations Check if part of domain Check if part of domain 33

Exponential and Logarithmic Equations Base should always be positive Check if part of domain 34

Exponential and Logarithmic Equations Check if part of domain Check if part of domain 35

Exponential and Logarithmic Equations Check if part of domain 36

Exponential and Logarithmic Equations Check if part of domain 37

Exponential and Logarithmic Inequalities 38

Exponential and Logarithmic Inequalities 39

Exponential and Logarithmic Inequalities 1/2 3 40

Exponential and Logarithmic Inequalities 41

Zeros: x = 4 x = -6 Interval -6 3 4 -6 3 4 -6 S.Y.Tan Signs change before and after zeros Interval 3 -6 3 4 -6 S.Y.Tan

Exponential and Logarithmic Inequalities 43

Zeros: x = 5/8 x = -2 Interval -2 1/3 5/8 -2 1/3 5/8 -2 S.Y.Tan Signs change before and after zeros Interval 1/3 -2 1/3 5/8 -2 S.Y.Tan