Download Logarithms: The Mystery Explained

(Created Tuesday, December 19, 2000. Edited Wednesday, December 20, 2000.)
(Last edited 14 December 2001.)
Created for LSCHS Mathletes web page (http://www.angelfire.com/pa3/lschsmathletes). Please do not
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Logarithms: The Mystery Explained
As you may remember, inverse functions are two functions whose operations undo each
other and whose composition is the function y=x (for example, the functions y=x3+4 and
y= 3 x  4 .)
The exponential function is the function y=ax, where a>0 and a  1. This function is
useful for various reasons, but its inverse is not easily intelligible. That’s where
logarithms come in.
The logarithm function is the inverse of the exponential function:
If y=ax, then x=loga y. a is termed the base of the logarithm.
For example, because 8=23, 3=log2 8. And, because the exponential and the logarithm
are inverses, a
log a x
= loga ax = x.
The range of the exponential is all positive numbers; the domain is all Reals.
The domain of the logarithm function is all positive numbers; the range
is all Reals.
Common Bases
The two most frequently used bases for logs are 10 and e.




The log base 10 of a number, the common log, is usually written log x; the 10 is
understood.
Examples:
 log 10000=4
 log .001=-3
 log 1=0
The log base e of a number, the natural log, is written ln x.
e  2.71828. If you really care where this number comes from:
1 1 1 1 1 1 1 1
 e          ... (remember, 0!=1)
0! 1! 2! 3! 4! 5! 6! 7!
 And, for you calculus addicts, e is defined in these ways:


1
e  lim (1  ) x , and
x
x
e1
 x dx  1.
1
The Laws of Logarithms
loga (xy)=loga x+loga y
x
loga ( )=loga x-loga y
y
loga (xn)=n loga x
logb x=
log a x
, where a and b are any valid base
log a b
(The base change formula)
loga ax=x
a
log a x
=x
loga a=1
loga 1=0 for any valid base a
Logs on the Calculator
Often, problems will ask for the values of expressions like log 6 78. Using the base
change formula,
log 6 78=
log 78 ln 78 log a 78
=
=
, where a is any valid base
ln 6
log 6
log a 6
So, entering “ln 78/ln 6” should yield the answer, as should entering “log 78/log 6”.
Practice problems
Without a calculator:
Given log 2=.3010, what is log 2 50?
Solve for x: 2log 6 5 = log 6 x + 2 (Keep in fractional form.)
Enjoy!