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Geology 261
Fall 2003
Mathematical Techniques
Exponential notation: We have already run across this in the expression of very large
and very small numbers, as in scientific notation. It is important to be able to correctly
and easily manipulate such numbers. The following is a brief summary of exponents.
An integer exponent, such as 34 (three to the fourth power) means that the number is
multiplied by itself the number of times indicated by the value of the exponent; in this
case 3x3x3x3 = 81.
An exponent of 1 does not change the value of the number.
x1 = x
An exponent of 0 is a special case.
x0 = 1
A negative exponent is the inverse of the positive exponent.
x-3 = 1/x3
So, for example, the concentration of a particular gas in the atmosphere might be 400
ppm which literally means;
400
 400 106
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Algebraic manipulations of exponents are straight forward;
400ppm 
x a  xb  x
 x  y
a
a b 
 xa  y a
xa
a b
  x  y
b
y
Most pocket calculators can handle exponents. Spread sheets also are able to do this.
Exponents do not have to be whole numbers. For example;
32.45  14.75527
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Calculators can easily take care of these non-integral values.
If the number is 1, then the number
1109
can also be written as
109
It is important to practice with your calculator so that you don’t waste time during an
examination and that you will get the correct answer.
Logarithms are extremely useful in that they reduce very large or very small numbers to
much smaller numbers. A particularly simple example is the logarithm to base 10, the
scheme we will be using for the rest of the semester. All the following items are valid for
base 10 logarithms (sometimes written as log10).
If x = 100 = (102) then
log x = log 100 = log 102 = 2 log 10 = 2
because by definition
log10 10 = 1.
similarly,
if x = 32.45 (see above) then
log x = log 32.45 = log 14.75527 = 1.168947
we could also write
log x = log 32.45 = 2.45 log 3 = 1.168947
Finally, we know that 101.16897 = 14.75527
How about negative exponents? Consider
x = 10-4 and log x = log 10-4 = -4 log 10 = -4
Note that the logarithm of 0 is undefined (does not exist).
Arithmetic operations with logarithms are done differently because they represent
exponents.
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log x + log y = log (xy)
log x – log y = log (x/y)
log xa = a log x
log 10a = a
log 1/a = log 1 - log a = -log a (log 1 = 0)
if log (6.022  1023) = 23.7797 then
1023.7797 = 6.022  1023
and finally,
10log a  a , that is, exponents and logarithms reverse each other.
For many of the equilibrium constants we will be using in the future, instead of reporting
values of K (the equilibrium constant), the value of – log K is listed. This is called the
pK. So if K = 69.19;
pK = - log 69.19 = -1.840
if pK = 68.6
then – log K = 68.6
log K = -68.6
10log K = 10-68.6
so that K = 10-68.6 = 2.511  10-69
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