Precalculus
Student Notes Chapter 3
Exponential/Logarithmic Functions
Name ________________________________________________
Hopefully you remember these properties of exponents:
EXPONENTIAL FUNCTIONS
Definition:
An exponential function has the form y  a x , where the base a is any positive real number other
than 1.
If a  1 and x is a positive exponent, the function models exponential growth.
If 0  a  1 and x is a positive exponent, the function models exponential decay.
If a  1 and x is a negative exponent, the function models exponential decay.
If 0  a  1 and x is a negative exponent, the function models exponential growth.
PROPERTIES OF THE PARENT EXPONENTIAL FUNCTION:
Domain: _______________________________
Range: ____________________________________
X-intercept: ____________________________
Y-intercept: ________________________________
Horizontal Asymptote: _____________________
(Additional info: Exponential functions are either always increasing, or always decreasing, over its domain.)
NATURAL BASE “e”:
Base “e” is an irrational base that is approximated by the number 2.71828
BASIC EQUATION FOR EXPONENTIAL GROWTH:
A  P(1  r ) t
BASIC EQUATION FOR EXPONENTIAL DECAY:
A  P(1  r ) t
EQUATION FOR COMPOUND INTEREST:
r

A  P 1  
n

A  Pe rt
EQUATION FOR CONTINOUSLY COMPOUNDED INTEREST:
EQUATION FOR HALF-LIFE:
1
A  P 
2
nt
t /T
LOGARITHMIC FUNCTIONS
If a > 0 and a  1, then the exponential function f(x) = ax is either increasing or decreasing. Thus, f is one-to-one
and hence has an inverse function f -1. This inverse function is known as the logarithm function with base a. Since
the domain of f(x) = ax is the set of all real numbers and the range is the set of all positive numbers, we can
interchange these to obtain the domain and range of f –1(x).
Domain of f(x) = ax : All reals
Domain of f –1(x) = log a x : x > 0
Range of f(x) = ax : y > 0
Range of f –1(x) = log a x : All reals
Definition of a “base-a” Logarithm
The logarithm with base a of a positive number x, denoted by log a x, is defined by
log a x = k if and only if x = ak, where a > 0, a = 1, and k is a real number.
The function, given by f(x) = log a x, is called the logarithmic function with base a.
For example,
log 2 8 = 3 since 23 = 8
log 5 625 = 4 since 54 = 625
Common Logarithms
We use LOG (or log without a base given) to represent log 10 . LOG is referred to as the Common Logarithm. On
your calculator you have a LOG key. This is the common log.
Natural Logarithms
We use LN to represent log e . The number e is referred to as the natural number named after Leonhard Euler.
The number is irrational and is approximately equal to 2.718281828.
LN (or ln) is referred to as the Natural Logarithm. On your calculator you have a LN key. This is the natural log.
PROPERTIES OF THE PARENT LOGARITHMIC FUNCTION:
Domain: _______________________________
Range: ____________________________________
X-intercept: ____________________________
Y-intercept: ________________________________
Vertical Asymptote:
_____________________
(Additional info: Logarithmic functions are either always increasing, or always decreasing, over its domain.)
Properties of logarithms:
log a 1 = 0 since a0 = 1 for a > 0.
log a a = 1 since a1 = a for a > 0.
log a (mn) = log a m + log a n
Proof:
Let x = log a m and y = log a n;
Then writing the corresponding exponential equations we have:
m = ax and n = ay ;
mn = (ax )(ay ) = ax + y ;
log a mn = log a ax + y = x + y = log a m + log a n.
log a (m/n) = log a m - log a n
Proof: Let x = log a m and y = log a n;
Then writing the corresponding exponential equations we have:
m = ax and n = ay ;
m/n = (ax )/(ay ) = ax - y ;
log a m/n = log a ax - y = x - y = log a m - log a n.
log a (mr) = r log a m
Proof:
Let x = log a m;
Then writing the corresponding exponential equations we have:
m = ax and mr = (ax )r = arx ;
log a mr = log a arx = rx = r log a m.
log a (ar) = r log a a = r * 1 = r