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