Pre-Cal 3.1 Exponential Functions

-Exponential Function: a function that can be written in the form:

-Review Laws of Exponents:

-Exponential Graphs: y =

-Transforming exponential graphs:

1 2 2 2 3 31. ( ) . ( ) . ( )g x h x k x ex x x

-natural base e: = 2.71828…

4 5 6 32. ( ) . ( ) . ( )g x e h x e k x ex x x

-To solve an exponential equation:

1. Rewrite the powers so both sides have a common base

2. Set exponents equal and solve for x

Precal 3.2 Logarithmic Functions

-Logarithmic Function: a function that can be written in the form:

f(x) = log a x

Note: f(x) = log a x and f(x) = ax are inverse functions

-Converting between exponential and logarithmic form:

y = log a x if and only if ay = x

Remember: Logarithms are Exponents!!

-Evaluating Logarithms: 1. Set the log equal to x 2. Rewrite in exponential form to solve

-Natural Logarithm: -Common Logarithm:

-Graphs of Logarithmic Functions:

1. Basic graph: y = log a x

log ln ( )

log log ( )e x x base e

x x base

10 10

( , )1 0

Precal 3.3 Properties of Logarithms-Basic Properties:

1. log a 1 = 0 2. log a a = 1 3. log a ax = x

-More Properties:

4. Product Rule:

5. Quotient Rule:

6. Power Rule:

log ( ) log logb b bMN M N

log log logb b b

M

NM N

log logbr

bM r M

-Change of Base Formula:

loglog

logba

a

MM

b

(if we use a = e)ln

ln

M

b

(if we use a = 10)log

log

M

b

Precal 3.4 Logarithmic Equations

-To solve a logarithmic equation:

1. Isolate logarithm on one side

2. Rewrite in exponential form

3. Solve for the variable

4. For base e equations, take the ln of both sides **calculator**

Precal 3.5 Exponential Growth & Decay

f x a ekx( ) -Given an exponential function:

1. If a > 0 and k > 0, f(x) is a growth function 2. If a > 0 and k < 0, f(x) is a decay function

-Growth & Decay Models:

-Exponential Population Model:

P0 = initial pop. r = percentage rate change t = time

P(t) = growth if r > 0, decay if r < 0

- Radioactive Decay Model:

R(t) = R0 = initial amounth = half-lifet = time

- Atmospheric Pressure Model:

P(h) = pressure at sea level = 14.7 lbs/in2

h = miles above sea level

Precal 3.6 Basic Combinations and Permutations

-Multiplication Principle of Counting:

If a procedure P has a sequence of S stages that can occur in R ways, then the number of ways that the procedure P can occur is the product of the R ways S stages can occur

-Combinations: the unordered selection of objects from a set.

- Permutations : the ways that a set of n objects can be arranged in order.

-The number of combinations of n objects taken r at a time is:

-The number of permutations of n objects taken r at a time is:

- n factorial = n! =special case: 0! = 1

n n n ( ) ( )...1 2 4 3 2 1

Precal 3.7 Expanding Binomials-The symbol:

-The formula:

The Binomial Theorem:

-Pascal’s Triangle: