Math 1314 – College Algebra
Section 6.3-6.4 Logarithmic Functions/Graphs of Logarithmic Functions
Are exponential functions one-to-one?
If b > 0 and b 6= 1, the logarithmic function with base b is
if and only if
logb x = y
Properties of Exponential Functions f (x) = bx :
Domain:
Range:
-intercept: ( , )
asymptote at
Properties of Logarithmic Functions f (x) = logb x:
Domain:
Range:
-intercept: ( , )
asymptote at
Passes through the point (1, )
Passes through the point ( , 1)
If b > 1,
If b > 1,
If 0 < b < 1,
If 0 < b < 1,
Ex: Find (a) log (−13)
(b) log (0)
Math 1314 OS
Section 6.3-6.4 Continued
Ex: Graph y = log2 x.
Ex: Find m:
(a) log3 9 = m
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(b) log8
1
=m
64
(c) log4 1 = m
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(d) logm
1
= −2
16
1
2
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(f) log5 m = 2
(e) logm 3 =
Domain:
Base 10 logarithms: The logarithm with base 10 is the common logarithm. log x means log10 x.
1
1
log 100
= −2 because 10−2 = 100
1
1
log 10
= −1 because 10−1 = 10
0
log 1 = 0 because 10 = 1
log 10 = 1 because 101 = 10
log 100 = 2 because 102 = 100
log 1000 = 3 because 103 = 1000
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Math 1314 OS
Section 6.3-6.4 Continued
Ex: Find x to four decimal places: log x = 0.7482
The logarithm with base e is the natural logarithm. ln x = loge x
Ex: Find (a) ln 17.32
(b) ln (log 0.05)
Ex: Solve (a) ln x = 1.335
(b) ln x = log 5.5
NOTE:
e(ln (?)) = ?
10(log (?)) = ?
ln e(?) = ?
log 10(?) = ?
We will discuss this in more detail in the next section.
Ex: Simplify: (a) e(ln (17.5)) − 11
(b) log 10(13) + 6
Ex: Graph and find the domain:
(a) y = ln x
(b) y = log x
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Math 1314 OS
Section 6.3-6.4 Continued
Ex: Find the domain, range, and vertical asymptote:
(a) h(x) = log2 (x + 2) + 5
(b) g(x) = log (12 − 2x)
x+4
(c) j(x) = ln
x−4
Ex: Find the domain: f (x) = log3 (5 − x) + log3 (2 + x)
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Math 1314 OS
Section 6.3-6.4 Continued
Ex: Find the domain: k(x) =
p
3
log3 x
Ex: Find the domain: k(x) =
p
4
log3 x
We can also shift and reflect graphs of f (x) = logb x.
Ex: Given f (x) = log3 (x + 2) + 4, find its inverse. Then graph both.
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