Page 2 - Calculus for the Clueless, Calc II
P. 2
Chapter 1—
Logarithms
Most of you, at this point in your mathematical journey, have not seen logs for at least a year, maybe longer.
The normal high school course emphasizes the wrong areas. You spend most of the time doing endless
calculations, none of which you need here. By the year 2000, students will do almost no log calculations due to
calculators. In case you feel tortured, just remember you only spent weeks on log calculations. I spent months!!!
The Basic Laws of Logs
2
y
1. Defined, log b x = y (log of x to the base b is y) if b = x; log 5 25 = 2 because 5 = 25.
1/2
2. What can the base b be? It can't be negative, such as -2, since (-2) is imaginary. It can't be 0, since 0 is
n
n
either equal to 0 if n is positive or undefined if n is 0 or negative. b also can't be 1 since 1 always = 1.
Therefore b can be any positive number except 1.
Note
1/2
1/2
The base can be 2 , but it won't do you any good because there are no 2 tables. The two most common bases
are 10, because we have 10 fingers, and e, a number that occurs a lot in mathematics starting now.
A. e equals approximately 2.7.
B. What is e more exactly? On a calculator press 1, inv, ln.
C. log = log 10
D. ln = log e (In is the natural logarithm).
3. A log y is an exponent, and exponents can be positive, negative, and zero. The range is all real numbers.
4. Since the base is positive, whether the exponent is positive, zero, or negative, the answer is positive. The
domain, therefore, is positive numbers.
Note
In order to avoid getting too technical, most books write log |x|, thereby excluding only x=0.
5. log b x + log b y = log b xy; log 2 + log 3 = log 6.
6. log b x - log b y = log b (x/y); log 7 - log 3 = log (7/3).
7
p
7. log b x = p log b x; ln 6 = 7 ln 6 is OK.
Note
Laws 5, 6, and 7 are most important. If you can simplify using these laws, about half the battle
(the easy half) is done.
Example 1—
Write the following as simpler logs with no exponents:
