B454EB9Ad01 - HANDOUT - GROWTH RATES AND LOGARITHMS I. What...

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HANDOUT - GROWTH RATES AND LOGARITHMS I. What is a natural logarithm (log)? Let’s begin with e . e is a number, approximately equal to 2.7. This number has some special properties and is often used in growth rate calculations. The exponential function is simply raising e to an exponent. For example, exp(2) is e 2 , or the square of e . The logarithmic function is the inverse of the exponential function. The log of a number is the exponent x that you would have to raise e to so that e x is equal to the argument of the function. For example, log(10) = 2.3 because e 2.3 = 10, or equivalently, exp(2.3) = 10. II. Rules for working with natural logarithms (logs): 1. The log of a product is the sum of the logs. log(AB) = log(A) + log(B) 2. The log of an exponential is multiplicative. log(C D ) = D * log(C) 3. Growth rate rule. A growth rate (as opposed to a percentage change, often used to approximate a growth rate) is defined as the difference between the log of the variable tomorrow and the log of the variable today. g Y = log(Y t+1 ) – log(Y t ) III. Why use log differences instead of percentage changes? A log growth rate is a continuous rate of growth. It is a rate of continuous compounding. A percentage change is specific to a particular time horizon. When the time horizons involved are short, a percentage change and a log difference often give very similar answers. However, when we study processes that evolve over many years or decades, like
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This note was uploaded on 12/10/2010 for the course ECONOMICS 432 taught by Professor Jannett during the Spring '10 term at Brown.

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B454EB9Ad01 - HANDOUT - GROWTH RATES AND LOGARITHMS I. What...

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