Notes on logarithms

# Notes on logarithms - Notes on logarithms Ron Michener...

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1 Notes on logarithms Ron Michener Revised January 2003 In applied work in economics, it is often the case that statistical work is done using the logarithms of variables, rather than the raw variables themselves. This seems mysterious when one first encounters it, but there are good reasons for it. Reviewing some basics. Since many of you have not seen logs in many years, let me review a few basic facts about them. If you have an expression such as y x a = y is said to be the “base a logarithm of x.” Practically speaking, there are only two bases that are ever used – base 10 (which is the one most commonly taught in high school) and base e . ( 2.718 e is a natural constant. Like π it is an infinite decimal.) A base e log is also known as a “natural log,” and it is the most commonly used log in advanced applications (including economics). Here are some examples: 2 100 10 = so that two is the base 10 log of 100. 1.386 4 e = so that 1.386 is the natural log of 4. Since natural logs are most commonly used in economics, I will use them exclusively. When I say log , I will mean the natural log. One useful property of logarithms is that they simplify certain arithmetic calculations. This used to be very important before the days of pocket calculators. For example, consider the product of two numbers, 1 x and 2 x . () 12 yy zx x e e e + == = As you can see, 1 y is the log of 1 x , and 2 y is the log of 2 x . The log of the product, z , is the sum + . Multiplication in the original numbers becomes addition in the logs, an easier operation. Similarly, if one raises x to the power p 11 p yp y p zx e e = the log of the result is just p times the log of x . Exponentiation in the original numbers becomes multiplication in logs.

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2 Why do natural logs appear in economics? Natural logs appear in economics for several reasons. First : If you have a variable – be it sales, population, prices, or whatever – that grows at a constant percentage rate, the log of that variable will grow as a linear function of time. The demonstration requires some calculus. Constant percentage growth means that dy dt a y = . The numerator on the left-hand side is the rate of growth of the variable y ; by dividing by y the left-hand side becomes the percentage rate of growth, which is equal to a constant, a . To solve this equation, we must rearrange and integrate. () log dy dt a y dy adt y dy y yc a t = = = =+ ∫∫ where the constant c is a constant of integration. What happens to y itself (as opposed to log y ) can be found by exponentiating, which undoes the log function. log * log y ca t t a t a t ee ye e c e + = =≡ To summarize what this derivation shows: If a variable grows at a constant percentage rate, the log of that variable will be a linear function of time, and the coefficient of time will be the percentage growth rate. The variable itself will exhibit exponential growth.
3 An Important Example: US GNP Here is a plot of US real gross national product against time, for 1896 – 1974.

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Notes on logarithms - Notes on logarithms Ron Michener...

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