LogarithmsA logarithm is the exponent given to a reference number (called the base) in order to write a number in ‘shorthand’. For the base 10 the logarithm of 100 is 2 and the logarithm of a million is 6. Another way to say this is 2 is the logarithm to the base 10 of 100. Logarithm to the base 10 is abbreviated log. Numbers between 10 and 100 have base 10 logarithms between 1 and 2. Numbers between 0 and 1 have base 10 logarithms between 0 and 1. Because it is not always easy to figure out the exponents for numbers, these have been worked out and are given in tables. Calculators also are programmed to calculate logarithms. What is the logarithm of 4? (Answer = .60206) In other words 10^{.60206} = 4. What number has log = .81291? (answer 6.5) In other words 10^{.81291} = 6.5 An advantage of using logarithms is that they are exponents and, as stated above:
You might wonder why bother when you can use a calculator. The fact is that very large numbers or very small numbers are conveniently expressed as logarithms. These can then easily be multiplied and divided using the rules above.
Natural logarithms Another commonly used base is e which is 2.7182818….. The exponents of e are called natural logarithms. Logarithm to be base e is abbreviated ln (for natural logarithm). So ln2 = e^{2} = 7.389… The number e is very interesting. It is the value reached in a unit time by a unit quantity that grows at a rate equal to itself. To understand this, let us assume you have $1.00 (unit quantity) and it is invested in a program that lets it grow at a rate equal to its value for a year (unit time). After a short time the dollar will have gained value. As a simple explanation: after 0.1 year it will have earned $.10 and will have become $1.10. At this time it is growing at a rate of $1.10 per year. After .2 year it will be worth $1.10 + .11 = $1.21 and will be growing at a rate of $1.21 per year. At the end of one year your $1.00 will be worth $2.718… (= e). This behavior can be expressed in an equation: $ = $_{o}e^{kt} Where $_{o} = starting dollars, $ = final dollars, k = the rate, t = the time. In the example above the rate was one dollar per dollar per year or k = 1. So if you start with one dollar ($_{o}) then at the end of one year (t = 1) you have 1 x e^{1} = e = 2.718. If you started with $2 then at the end of one year you have 2 x e^{1} = 2e = 5.436. If you started with $1 but the rate was $2 per dollar then after one year you have 1 x e^{2} = 7.389. After two years you have 1 x e^{2x2} = 1 x e^{4} = 54.598.. One natural system that appears to show exponential growth is population. Exponential decay is simply a quantity that decreases at a rate equal to itself. This behavior is shown by radioactive decay. The fewer the atoms the more slowly the number decreases. The decay constant for an isotope is -l (the rate of decay). The minus sign shows the number will get smaller. The equation is N = N_{o}e^{-l t}. N_{o} is the starting number of radioactive atoms and N is the number left after time t, when the rate of decay (decay constant) is -l . |