5.4) Logarithmic Functions:
Recall the graph of the exponential function, f (x) = ax, where a > 0 and a ≠ 1 is
either an increasing function (__________) or a decreasing function (__________).
Since y =f(x) = ax, a > 0, a≠ 1 is ______________, the it must have an ________________ that
is defined implicitly (not solved for y by itself) as:
The inverse function is so important that it has its own given name which we call the
logarithmic function.
The logarithmic function to base a, where a > 0 an a ≠ 1, is written as y = loga (x). Customary to
read this as "y IS the logarithm to the base a of x." Defined as:
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Main Players:
Look at the interplay between all the players involved:
2  log 5  25 
 1 
log1/3    3
 27 
MEANS
MEANS
Example: Evaluate the following:
(a)
(c)
log 4  64 
log 4  8 
(b)
(d)
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log 
log 27  81
Logarithmic Function properties
For the domain of y = loga (x), we need x > 0. This makes sense because the Logarithmic
function is the inverse of the exponential function and we established in section 5.2:
 Domain of f -1(x) = Range of f(x)

Range of f -1(x) = Domain of f(x)
f  x  ax
f 1  x   log a  x 
Domain:
Domain:
Range:
Range:
y-intercept:
y-intercept:
x-intercept:
x-intercept:
Vertical Asymptote:
Vertical Asymptote:
Horizontal Asymptote:
Horizontal Asymptote:
Key Points:
Key Points:
Sketch:
f  x   log 3  x 
Sketch: f  x    log 3  x  3 
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Sketch:
f  x   log 3  3  x   2
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Common Logarithm:
Natural Logarithm:
Evaluate the following:
log
ln 4 e
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
5
1000

Domains
The domain of the logarithmic function,
requires:
f
x 
lo g a  x 
, a > 0 (positive base)
(Example) Find the domain of each of the following:
(a)
(c)
f  x   log 5  5  2 x 
f  x   ln  x 
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
(b) f  x   log x  4 x  32
2

Solving Logarithmic Equations
Looking for the value of the variable that makes the logarithmic equation a TRUE statement.
M  log a N
log a  x   N
If log a
Also, if
, then ___________________
, then ___________________
(Example) Solve the following equations for the unknown variable:
(a)
log 8  2 x  5   log 8  4 x  25 
log 5  3  4 x   3
(b)
(c)
(d)
 1 
log m    3
 27 
e 2 x  3  12
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Spreading Rumors
A model for the number N of people in a college community who have heard a certain rumor is
N  d   P 1  e 0.15 d 
where P is the total population of the community and d is the number of days that have elapsed
since the rumor began. In a community of 1000 students, how many days will elapse before
450 students have heard the rumor?
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