Logarithmic Functions

Significance

The logarithmic function f(x) = log_bx is the inverse function of the exponential function b^x:

log_b(b^x) = b^log_bx = x.
In other words, y = log_bx 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 log₂x. 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 log₂x.

The Algebra of Logarithms

The main rules for manipulating logarithms are

log_bxy = log_bx + log_by.
log_bx^y = y log_bx.
log_b1=0.
log_b0 is undefined (or -infinity).
log_bx = (log_bc)log_cx.

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_bx can be thought of as a scaled (along the y-axis) 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.