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Notes 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; 400 400 106 6 10 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 1 Calculators can easily take care of these non-integral values. If the number is 1, then the number 1109 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. 2 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 3