Logarithms: 2008
JamesS Page1 of 3
Logarithms: A Logarithm is really an Exponent. This is a fundamental idea to
keep in mind when using Logs. By definition:
Y = Log
B
(X) if and only if : X = B
Y
I use what I call the Log Loop to see this: Drawing a loop from the base (B) around through
the Y to the X and read it as: B to the Y = X (B
Y
=X).
Ex: Find X: => using the definition of a log…
therefore X = 5 XLog =)32(
2
322 =
X
A logarithm is a variation in the form of an exponential number. The two most commonly used
logarithms are Base 10 and Base 'e'. Log (A) is read: Log base 10 of A. Log base 10 (Log
10
) is
referred to as the “Common” logs, whereas Log base e (Log
e
) is referred to as the “Natural”
logs and uses the abbreviation (Ln). Unless otherwise indicated the term Log (x) is always
understood to be base 10 or Log
10
(x)
Log Terminology: Base, Expand, Compress, Exponentiate, Inverse,
Log Base: There are two primary bases that are used: Base 10 (Common Log) and
Base e (Natural Log). It is common practice to differentiate between them using the terms Log
and Ln. The graph of a Log in any base is essentially the same; the difference being the rate of
change along the curve of the graph, which means that the value obtained from the Log (A) vs
Ln (A) will be different.
Expand: Expanding a Log means going from a single Log of some value to two or
more Logs. This is easily understood when you look at the Multiplication
Property.
Log (A*B) = Log (A) + Log (B)
You begin with a single Log of (A times B) and then expand it to the sum of two individual
Logs:
Log (A) + Log (B)
We say that the original Log of (A*B) has been “expanded.” The purpose of expanding, besides
giving you practice in using the properties, is to allow these Logs to be further handled
algebraically. As an example of this lets look at: Log ( 37e
-kt
) By using the Multiplication
and Exponent
Property we can “expand” this Log to: Log (37) + (-kt)Log (e)
As you can see we now have a more simple algebraic statement; the exponent (-kt) has become a
simple Coefficient; of course, in reality we would have used the Natural log (Ln) for this Log
because of the “e” term:
Ln ( 37e
-kt
) = Ln (37) + (-kt)Ln (e)
Since Ln (37) is just a number and Ln(e) = 1 ; we have: 3.6 + (-kt)
Compress: Compressing is just going the opposite direction of Expanding; this
may be as simple and taking a Log Coefficient and moving it to the exponent position.
3 Log (A) = Log (A)
3