Definition:

Any number (X) can be expressed by any other number b (except zero) raised to a power x; which is, there is always a value of x such that X = b^x.  For example, if X = 8 and b = 2, x = 3.  For X = 8 and b = 4, 8 = 4^x is satisfied if x = 3/2.

4 ^3/2 = (43)^1/2 = (64)^1/2 = 8

Or

4 ^3/2 = (4^1/2)³ = 2³ = 8

In the equation X = b^x, the exponent x is the logarithm of X to the base b.  Begin in equation form, x = log_b  X, that reads x is the logarithm to the base b of X. In common terms, the logarithm of a number to a base b is the power to that base b must be raised to yield the number.  The rules for logs are a direct consequence of the rules for exponents, because that is what logs are. In  multiplication,  for  instance,  consider  the  product  of  two  numbers  X  and  Y. Expressing each as b raised to a power and using the rules for exponents:

XY =(b^x)(b^y) = b^x+y

Presently, equating the log_b of the first and last terms, logb  XY = log_b  b ^x+y.

Since the exponent of the base b (x+y) is the logarithm to the base b, Log_b  b^x+y = x+y.

log_b  XY = x+y

Similarly, since X = b^x  and Y = b^y, log_b  X = x and log_b XY = log_b  Y = y. Substituting these into the log previous equation,

log_b XY = log_b X + log_b Y

Before the advent of hand-held calculators it was general 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 that had this log.   This is the product of the two numbers. A  slide  rule  is  designed  to  add  logarithms  as  numbers  are multiplied.

Logarithms can simply be computed with the calculator using the keys identified previously.