Logarithms Properties and Uses Some background A logarithm
Logarithms Properties and Uses
Some background • 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 102 = 100. So the log of 100 to the base 10 equals 2. – 10 * 10 = 1000. So 10 cubed equals 1000 or 103= 1000. So the log of 1000 to the base 10 equals 3. Etc…. .
Base 10 Logarithm Examples CASE NUMBER BASE 10 LOG 1. 000 0. 000 2. 000 10. 000 1. 000 3. 000 100. 000 2. 000 4. 000 200. 000 2. 301 5. 000 500. 000 2. 699 6. 000 1000. 000 3. 000 7. 000 5000. 000 3. 699 8. 000 10000. 000 4. 000 9. 000 15000. 000 4. 176 10. 000 30000. 000 4. 477
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 VAR 00008 LOGVAR 1 1. 000 0. 000 2 2. 000 10. 000 1. 000 3 3. 000 100. 000 2. 000 4 4. 000 200. 000 2. 301 5 5. 000 500. 000 2. 699 6 6. 000 1000. 000 3. 000 7 7. 000 5000. 000 3. 699 8 8. 000 10000. 000 4. 000 9 9. 000 15000. 000 4. 176 10 10. 000 30000. 000 4. 477
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 2. 718145. [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.
Antilog or exponentiating function Natural log 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 = 100 + 2 * 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 $104.
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 +. 02*100 = 102 Year 2: F = 102 +. 02 (102) = 104. 04 Year 3: F = 104. 04 +. 02 (104. 04) = 106. 12
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
Example of Interest Calculations: $100 invested at 2% and 6% over 10 time periods 1 2 3 4 5 6 7 8 9 10 SIMPLE 102. 000 104. 000 106. 000 108. 000 110. 000 112. 000 114. 000 116. 000 118. 000 120. 000 COMPND 102. 000 104. 040 106. 120 108. 240 110. 410 112. 620 114. 870 117. 170 119. 510 121. 900 TIME 1. 000 2. 000 3. 000 4. 000 5. 000 6. 000 7. 000 8. 000 9. 000 10. 000 LN COMPND 4. 625 4. 645 4. 665 4. 684 4. 704 4. 724 4. 744 4. 764 4. 783 4. 803 LOG COMPND 2. 009 2. 017 2. 026 2. 034 2. 043 2. 052 2. 060 2. 069 2. 077 2. 086 SIMPLE 6 106. 000 112. 000 118. 000 124. 000 130. 000 136. 000 142. 000 148. 000 154. 000 160. 000 COMPND 6 106. 000 112. 360 119. 100 126. 250 133. 820 141. 850 150. 360 159. 380 168. 950 179. 080
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