Swing the Logarithmic Curve around (1, 0)

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The logarithmic function to the base , where
and
, is defined by
if and only if
; the domain is
and the range is
.
Contributed by: Abraham Gadalla (March 2011)
Open content licensed under CC BY-NC-SA
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When considering the common logarithm (i.e., base 10), we notice that as the values decrease from 1 to 0, the curve falls rapidly, and for
, it approaches the negative
axis asymptotically. As the
values increase from 1 to 10, the function increases monotonically from 0 to 1, and as
values increase by a factor of 10 (for example, from 10 to 100) the function increases from 1 to 2. The same applies for the intervals
,
, and so on. Because the changes are very small for such large intervals, the curve can be well approximated by a straight line.
To switch bases, we let ; we will show that
.
By definition, implies
.
Taking the to the base
of both sides gives
.
Dividing by gives
. Replacing
by
yields
.
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