Chapter 5 · Chapter 5 Exponential and Logarithmic Functions. Section 3 ... Domain: all real...

Chapter 5Exponential and Logarithmic

Functions

Section 3Exponential Functions

A function in the form f(x) = Cax is an exponential function where “a” is a positive real number (a ≠ 1, C ≠ 0). [a is the base, C is the initial value]

Domain: all real numbers

These functions show that as x increases by one, y would increase by a multiple of “a”

Example 2: Identifying Linear or Exponential Functions

Identify whether the following functions are exponential or not. If so, what is “a”?

X f(x) x f(x)-1 3 -1 ¼0 6 0 11 12 1 42 18 2 163 30 3 64

Laws of Exponents (page 270)

as x at = as + t

(as)t = ast

(ab)s = as x bs

1s = 1

a-s = (1/as) = (1/a)s

10 = 1*any # to 0 power = 1

Graphing using transformations if basic function is exponential:

Basic: f(x) = ax where a = positive numbers only

(-1, 1/a) (0, 1) (1, a)

H.A.: y = 0

Domain: {x| all real numbers}

Range: {y| y > 0} based on horizontal asymptote and where graph is located

Example:Graph and find the domain and range of the following:f(x) = 3x+1 – 2

Basic:

Example:Graph and find the domain and range of the following:f(x) = - 1/2x + 1

Basic:

Solving Exponential Equations:

There needs to be ONE base raise to some power equal to that same base raised to some power

au = av

Then it can be said that u = v

*use the law of exponents to simplify

Example 7:

Solve for x 3x+1 = 81

Example:

Solve for x 8 –x + 14 = 16x

Example 8:

Solve for x 𝑒−𝑥2= (𝑒𝑥)2 ×

1

𝑒3

EXIT SLIP