EXPONENTS AND LOGARITHMS 



APPENDIX D 



*rfiJSiS«^ 



Multiplication 



103 X 103 ^ 106 (1000 X 1000 = 1 million), 



.-. log 103 + log 103 (2 X log 103) = 3 _|. 3 = 6 

 (i.e., multiplying numbers = adding exponents 

 or logs). 



Division 



/1, 000,000 



106 ^ 102 = 104 ^ : — = 10,000 



V 100 



)• 



.". in logs: 6 — 2 = 4 (i.e., dividing numbers 

 subtracting exponents or logs). 



Make a graph, plotting these numbers against 

 their logs. From this you can interpolate inter- 

 mediate numbers. From your graph read the 

 logs of 2.35, 9.76, 3.87. 



The log of 



5.76 = 0.7604, 



576 = 2.7604, 



576 million = 8.7604, 

 0.576 = T.7604, 



0.00576 = 3.7604. 



The number before the decimal point is called 

 the characteristic: 



characteristic 43210 T2345 

 number 00000.00000 



To multiply decimals: 



0.003 X 0.090 = 0.00027 



logs: 3.48 + 2.95 = 4.43 



Check with your graph. 



An exercise in logarithms: the pH scale 



In pure water or in any aqueous solution, 

 whatever its alkalinity or acidity, the product of 

 the concentrations of hydrogen and hydroxyl 

 ions in moles per liter is lO-i*: 



(H+)(OH-) = 10-14 



.-. log (H+) + log (0H-) = - 14 



or, changing signs, 



-log(H+) - log(OH-) = 14 



— log (H+) is called the pH (Sorensen). In a 

 neutral solution, (H+) = (OH-) = IQ-^ moles 

 per liter, i.e., 



log(H+) = -7 



.-. pH = 7 



149 



