


Logarithms
Logarithms are closely related to exponents and roots.
A logarithm is the power to which you must raise a given number,
called the base, to equal another number. For example, log_{2} 8
= 3 because 2^{3} = 8. In this case, 2 is
the base and 3 is the logarithm.
The Math IIC likes to use logarithms in algebra problems,
mostly in simple equation solving problems (which we will cover
in the next chapter). For these types of questions, the key is remembering
that a logarithm problem is really an exponent problem. Keeping this
in mind should help reduce the mystery that seems to surround logarithms.
In fact, once you get the hang of it, you’ll realize that solving
logarithmic equations is actually quite easy.
Having defined a logarithm in a sentence, let’s show it
symbolically. The three equations below are equivalent:
For example, log_{4} 16 = 2 because
4^{2} = 16 and = 4. You should be able to see now
why the three topics of exponents, roots, and logarithms are often
linked. Each provides a way to isolate one of the three variables
in these types of equations. In the example above, a is the
base, b is the exponent, and x is
the product. Respectively, finding the root, logarithm, and exponent
isolates these values.
Logarithms and Calculators
Unless the logarithm is a very simple one, you won’t be
able to mentally calculate it—so the calculator becomes an important
tool. But there is one critical thing you need to be aware of. On
your calculator, the “LOG” button assumes a base of ten. This means
that for the equation log_{4} 16 = 2, if you
punched in LOG 16, you would get log_{10} 16.
Some calculators can calculate a logarithm with any base,
but less advanced calculators might not. When you need to calculate
a logarithm with a base other than 10 and your calculator does not
have the capability, keep in mind that: log_{b} a =
log_{c} a ⁄
log_{c} b.
Calculate a few logarithms for practice:
Operations on Logarithms
You will rarely see a test question involving basic logarithms
such as log_{10} 100 or log_{2 }4.
In particular, on the logarithm questions you’ll see in the Algebra
chapter, you’ll need to be able to manipulate logarithms within
equations. You should therefore know how to perform the basic operations
on logarithms:
The Product Rule:
when logarithms of the same base are multiplied, the base
remains the same, and the exponents can be added:
The Power Rule:
when a logarithm is raised to a power, the exponent can
be brought in front and multiplied by the logarithm:
You might have noticed how similar these rules are to
those for exponents and roots. This similarity results from the
fact that logarithms are just another way to express an exponent.
The Quotient Rule:
when logarithms of the same base are divided, the exponents
must be subtracted:
Natural Logarithms
Natural logarithms appear on the Math IIC but not on the
Math IC test. A natural logarithm is one with a base of e.
The value e is a naturally occurring number, infinitely
long, that can be found in growth and decay models. The natural
logarithm will most likely be used in problems of growth and decay.
A common numerical approximation of e is
2.718, which you could easily discover by punching e^{1} into
your calculator. The symbol for a natural logarithm is ln, instead
of log. The following three equations are equivalent:
Working with natural logarithms is just like working with
logarithms; the only difference is that the base for natural logarithms
is always e. You might also be asked to identify
the graphs of ln x and e^{x}.
Take a good look at their general shapes:
