Download Exponential and Logarithmic Functions Honors Precalculus

Honors Precalculus Handout – Chapter 4 – M.C.
Exponential and Logarithmic Functions
DEFINITION: An exponential function is a function of the form
f(x) = ax where a is a positive number and a is not equal to one. (?)
A. Graphs of Exponential Functions:
1. f (x) = a x where a > 1
 Domain: All real numbers
 Range: (0, )
 Increasing Function
 Passes through points (0,1) and (1,a)
 End Behavior: As x   , f(x)  0 (but f(x)  0)
2. f (x) = a x where 0<a<1





Domain: All real numbers
Range: (0, )
Decreasing Function
Passes through points (0,1) and (1,a)
End Behavior: As x   , f(x)  0 (but f(x)  0)
What does the graph of f(x) = a –x look like? All transformation techniques also apply to
exponential functions.
B. Definition:
F1  1IJapproaches as x   .
The number e is defined as the number that the expression G
H nK
n
Note: The number e (approximately 2.72) is an irrational number named after its discoverer
Leonhard Euler.
C. Solving Exponential Equations using the Rules of Exponent and the following property:
a u  a v iff u  v
Examples: p. 298
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D. Logarithmic Function:
Note: The exponential equation y = a x, where a > 0 and a  1 is a one-to-one function and has an
inverse which is defined implicitly by the equation x = a y.
Characteristics of the Inverse Function:
 Domain of f = Range of f -1 = ,



-1
b g
=b
0,g
Range of f = Domain of f
y-axis is a vertical asymptote
The points (a,1) and (1,0) are points on the graph of f -1.
DEFINITION: The logarithmic function to the base a, where a > 0 and a  1 , is denoted by
y  loga x and is defined by y  loga x iff x  a y
Exponential Form:
Logarithmic Form:
3x=9
iff
log 3 9 = x
5 2x = 125
iff
log 5 125 = 2x
23=8
iff
log 2 8 = 3
x3=4
iff
log x 4 = 3
Examples: p. 310-311
Notes:
1. The log of a number is the exponent when written in exponential form.
2. If the base of a logarithmic function is the irrational number e, then we have the natural
logarithm function. This function is given a special symbol. That is y  ln x iff x  e y .
3. y  ln x and x  e y are inverse functions.
4. Logarithms to the base 10 are called common logarithms. This logarithm is abbreviated as
y = log x.
log x
5. Change of base formula: loga x 
log a
2
6. Properties of Logarithms:
a. log a 1 = 0, log a a = 1
b. alog a M = M; log a a r = r
c. log a (MN) = log a M + log a N
d. log a
M
 log a M – log a N
N
e. log a M r = r log a M
Examples: p. 321
Rules:
1. To find the exact value of a logarithm, write the logarithm in exponential form along with
the fact that a u  a v iff u  v .
2. Equations that contain logarithms are called logarithmic equations. In solving this type of
equation be sure to check each apparent solution in the original equation and discard any
that are extraneous.
Examples: p. 327
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