Logarithms Properties and Uses

Some background Read pp in Agresti and Finlay. A logarithm (generally called the “log” of a number) is the “power to which a given base must be raised to equal that number.” Common bases: Base 10 and Base e.

Examples Base 10 logarithms: –10 * 10 = 100. So 10 squared equals 100 or 10 2 = 100. So the log of 100 to the base 10 equals 2. –10 * 10 * 10 = So 10 cubed equals 1000 or 10 3 = So the log of 1000 to the base 10 equals 3. Etc…..

Base 10 Logarithm Examples CASE NUMBER BASE 10 LOG

Properties of Logarithms A proportionate change in the original number becomes an arithmetic change in the logarithm. The base 10 logarithm of 10 is 1. The base 10 log of 100 is 2. The base 100 log of 1000 is 3. The tenfold increase from 1 to 10, 10 to 100 and 100 to 1000, is a 1 unit increase in the base 10 log, from 1 to 2 to 3.

Properties of Logarithms The properties of logarithms make them useful for the analysis of growth (or decay). Economists, demographers and historians remeasure series which exhibit strong growth patterns in logarithms to capture the proportionate change as an arithmetic change. OLS regression can then be used to explore relationships.

Base 10 Logarithm Examples VAR00008 LOGVAR

Properties of Logarithms The most common base used is base e, or natural logarithm, which is also is written ln. “e” is a numeric constant (like pi) which formally is equal to the limit of the sequence of terms (1 + 1/N) N as the integer N grows larger and larger. An approximation when N = 10,000 is [that is (10,001/10,000) 10,000 ]

Log and Anti Log Transformations Scientific calculators and statistical programs have functions to transform a number into its log value. Log values can be converted back to the original value by using the anti log or the exponentiating function.

Natural log function Antilog or exponentiating function

Simple and Compound Interest Simple interest formula: y = a + b*x where –Y = final value –a = initial value –b = interest rate –x = time So, after 10 years, a $100 investment with simple interest at the rate of 2%: –Y = * 10 = $120

Example: Compound Interest Future value = Present Value * (1 + i) n Future value = Product of Present value times (1 plus the interest rate raised to the number of time periods). So if present value = $100, and the interest rate is 2% (.02) and the time period is one year, the future value after a year is $102. After two years, the future value is $102 * 1.02 or $

Compound Interest example Year 1: F = (P + r*P) Year 2: F = (P + r*P) + r (P + r*P) Year 3: F = ((P * r*P) + r (P + r*P)) + r* ((P + r*P) + r (P + r*P)) or Year 1: F = *100 = 102 Year 2: F = (102) = Year 3: F = (104.04) =

Compound Interest example Year 1: F = (P + r*P) Year 2: F = (P + r*P) + r (P + r*P) Year 3: F = ((P * r*P) + r (P + r*P)) + r* ((P + r*P) + r (P + r*P)) or Year 1: F = P (1 +r) Year 2: F = P (1 + 2 r + r 2 ) = P (r+1) 2 Year 3: F = P (r+1) 3

SIMPLE COMPND TIME LN COMPND LOG COMPND SIMPLE6 COMPND Example of Interest Calculations: $100 invested at 2% and 6% over 10 time periods