Any number (X) can be expressed by any other number b (except zero) raised to a power x; that
is, there is always a value of x such that X = bx. For example, if X = 8 and b = 2, x = 3. For
X = 8 and b = 4, 8 = 4x is satisfied if x = 3/2.
In the equation X = bx, the exponent x is the logarithm of X to the base b. Stated in equation
form, x = logb X, which reads x is the logarithm to the base b of X. In general terms, the
logarithm of a number to a base b is the power to which base b must be raised to yield the
number. The rules for logs are a direct consequence of the rules for exponents, since that is what
logs are. In multiplication, for example, consider the product of two numbers X and Y.
Expressing each as b raised to a power and using the rules for exponents:
XY = (bx) (by) = bx+y
Now, equating the logb of the first and last terms, logb XY = logb bx+y.
Since the exponent of the base b (x+y) is the logarithm to the base b, Logb bx+y = x+y.
logb XY = x+y
Similarily, since X = bx and Y = by, logb X = x and logb Y = y. Substituting these into the
logb XY = logb X + logb Y
Before the advent of hand-held calculators it was common to use logs for multiplication (and
division) of numbers having many significant figures. First, logs for the numbers to be
multiplied were obtained from tables. Then, the numbers were added, and this sum (logarithm
of the product) was used to locate in the tables the number which had this log. This is the
product of the two numbers. A slide rule is designed to add logarithms as numbers are
Logarithms can easily be computed with the calculator using the keys identified earlier.