102
Chapter 1 | Functions and Graphs
3 =8,
log 2 (8) =3
since
2
⎛ ⎝ 1
⎞ ⎠ =−2 since
10 −2 = 1
log 10
= 1 100 ,
100
10 2
b 0 = 1for any base b >0.
log b (1) =0
since
x are inverse functions,
Furthermore, since y = log b ( x ) and y = b
log b ( x )
log b ( b = x . The most commonly used logarithmic function is the function log e . Since this function uses natural e as its base, it is called the natural logarithm . Here we use the notation ln( x ) or ln x to mean log e ( x ). For example, ln( e ) = log e ( e ) =1, ln ⎛ ⎝ e 3 ⎞ ⎠ = log e ⎛ ⎝ e 3 ⎞ ⎠ = 3, ln(1) = log e (1) =0. Since the functions f ( x ) = e x and g ( x ) = ln( x ) are inverses of each other, ln( e x ) = x and e ln x = x , and their graphs are symmetric about the line y = x ( Figure 1.46 ). x ) = x and b
Figure 1.46 The functions y = e x and y = ln( x ) are inverses of each other, so their graphs are symmetric about the line y = x .
At this site (http://www.openstax.org/l/20_logscale) you can see an example of a base-10 logarithmic scale.
In general, for any base b >0, b ≠1, the function g ( x ) = log b ( x ) is symmetric about the line y = x with the function f ( x ) = b x . Using this fact and the graphs of the exponential functions, we graph functions log b for several values of b >1 ( Figure 1.47 ).
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