|A logarithm is an exponent.|
Note, the above is not a definition, merely a pithy description.
Just as subtraction is the inverse operation of addition, and taking a square root is the inverse operation of squaring, exponentiation and logarithms are inverse operations. Finding an antilog is the inverse operation of finding a log, so is another name for exponentiation. However, historically, this was done as a table lookup. Some history was given earlier and the formal definition is repeated below, this time with restrictions.
y = logbx if and only if by = x,|
where x > 0, b > 0, and b 1.
As noted above, the base can be any positive number (except 1). However, two choices are most usual: 10 and e=2.718281828.... Logs to the base 10 are often call common logs, whereas logs to the base e are often call natural logs. Logs to the bases of 10 and e are now both fairly standard on most calculators. Often when taking a log, the base is arbitrary and does not need to be specified. However, at other times it is necessary and must be assumed or specified.
At the high school level only,
log x consistantly means log10x.|
In college, especially in mathematics and physics, log x consistantly means logex.
A popular notation (despised by some) is: ln x means logex.
To calculate logs to other bases, the change of base rule below (#4) should be used. It is only multiplication by a constant (1 / logab).
These four basic properties all follow directly from the fact that logs are exponents. In words, the first three can be remembered as: The log of a product is equal to the sum of the logs of the factors. The log of a quotient is equal to the difference between the logs of the numerator and demoninator. The log of a power is equal to the power times the log of the base.
Additional properties, some obvious, some not so obvious are listed below for reference. Number 6 is called the reciprocal property.
From this we can readily verify such properties as: log 10 = log 2 + log 5 and log 4 = 2 log 2. These are true for either base. In fact, the useful result of 103 = 1000 1024 = 210 can be readily seen as 10 log10 2 3.
The slide rule below is presented in a disassembled state to facilitate cutting. (Also, by putting it below, it will be at the bottom of page 3 and have blank paper behind it.) The portion above slides in the center of the portion below and should be printed, then cut out for demonstration purposes as follows.
Following, is an interesting problem which ties the quadratic formula, logarithms, and exponents together very neatly.
Blank space so when printed with Mozilla (oops, no boxes) it is in back of the slide rule.
However x -3 since the domain of log is only the positive reals. (bx can never be a negative number with b > 0).
The next example (6.11#51) combines logarithms with simultaneous equations. It is also very convenient to introduce the concept of substitution, which is so useful in calculus.
Let u=log9x and v=log8y. By the reciprocal property above, 1/u=logx9 and 1/v=logy8.
We can rewrite our equations now as: