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Section 4.3 Logarithmic Functions
(ALERT: Based on 4.1 4.2)
Logarithms
1. Definition: y = log (^) a(x) (read as y is logarithm to the base a of x.)
y โ x =a for all
a > 0 , aโ 1. Plugging the first to the second, we have log (^) a(x) x = a. In another word,
intuitively log (^) a (x) gives the power a needs to be raise to to obtain x.
z Basic properties:
log (^) a 1 = 0 , because 1
0 a = ,
log (^) a a= 1 , because a =a 1 ,
a m m log (^) a ( )= , because m m a = a ,
a x a x =
log () , by definition (subtle but most important).
z Common log:
log x =log 10 x
z Natural log:
ln x = loge x, where e =2.71828 1828459045 23536....
Example 1
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Example 2
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Example 3
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Logarithmic Function and Exponential function
We know that y = log (^) a(x) y โ x =a , so if f ( x)= loga(x) then x g ( x)=a is the
inverse of f (x), i.e. x f x =g x =a โ ( ) ( ) 1 That is to say that the logarithmic function and
the exponential function with the same base are the inverse function of each other. Then
we recall from Section 4.2 the relation between a function and its inverse.
z The Domain of
x y = a , ( โโ, +โ), is the Range of y = log (^) a(x).
z The Range of x y = a , ( 0 ,+โ) , is the Domain of y = log (^) a(x).
z The two identities are more sensible if you view them as the composition of a function
and its inverse: a m
m log (^) a ( )= , a x a x =
log()
Exercise 4
y = log (^1) / 2 (x) and
x y (^) โ โ
are the inverse function of each other.
Properties: The graph of f ( x)= loga x, 0 <a< 1 / 2 looks similar to
f ( x)=log 1 / 2 (x )
z Strictly decreasing, i.e., if x 1 > x 2 , then 1 2
x x a < a.
z y โโ as x โ 0 , i.e. y=0 is a vertical asymptote.
z y โโโ as x โโ.
z The x-intercept is 1. z The graph contains the points (1,0) and (a,1).
z y = log (^1) / 2 (x) and
x y (^) โ โ
are the inverse function of each other.
Exercise 5
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