
Logarithmic Functions
Significance
The logarithmic function f(x) = log_{b}x is
the
inverse
function of the exponential function b^{x}:
log_{b}(b^{x}) = b^{logbx} = x. In other words, y = log_{b}x if and only if x = b^{y}. 
Such functions
are used in solving exponential
equations. Logarithmic functions
turn multiplication into addition. Before the advent of calculators,
first
tables of logarithms and then slide rules based on logarithms were used
as an aid to multiplication.
Standard Notation
There are three bases for logarithms which in widespread use.  The natural logarithm of x, written ln x, is the logarithm to the base e. Thus e^{x} and ln(x) are inverse functions. The natural logarithm is the one which has the nicest purely mathematical properties and is the one which we use almost exclusively in calculus.
 The common logarithm of x, often written simply as log x, is the logarithm to the base 10. Thus 10^{log x} = x.

In some disciplines  such as communications, computer science, and information theory  the "correct" base for the logarithm is base 2. For them, log(x) means log_{2}x. If x is an integer, this measures the number of bits it takes to write x. We will not use this notation in CalculusQuest^{TM}. 
CS TIP: The number of bits needed to write the positive integer x in base 2 is obtained by rounding UP the number log_{2}x. 
The Algebra of Logarithms
The
main rules
for manipulating logarithms are
 log_{b}xy = log_{b}x + log_{b}y.
 log_{b}x^{y} = y log_{b}x.
 log_{b}1=0.
 log_{b}0 is undefined (or infinity).
 log_{b}x = (log_{b}c)log_{c}x.
The last rule shows that all logarithmic functions are the same except
for a constant multiple.
Graphs Logarithmic Functions
The graph of the function y = log_{b}x can be thought of as a scaled (along the yaxis) version of the graph of y = ln(x).
All logarithmic functions log_{b>}x pass through the point (1,0) and have a vertical asymptote at zero. All have as their domain the set of positive real numbers (which is the range, or codomain, of every exponential function), and
all logarithmic functions increase as
x increases, albeit VERY SLOWLY! Logarithmic growth is slower than that of any rational function.
