In working with production functions and growth models, we often have towork with exponents, including fractional exponents.A brief review of the basics follows. Exponents Definitions xa = x times x times x ... to a total of a times.x -a = 1 / xa Negative exponents give the reciprocal of the positive expontneFor example x -2 = 1 / x2 Operations Multiplying variables raised to a power involves adding their exponents.
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xa times xb = x a + b x2 times x3 = x 5 Dividing variables raised to a power involves subtracting their exponents. xa divided by xb = x a - b x5 / x3 = x 2 Exponentiation of variables raised to a power involves multiplying the exponents.
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xa raised to the b power = x a b x5 squared = x 10 Note: There are no easy rules for addition and subtraction of variables raised to a power. Logarithms and percentage changes Logarithms are exponents and hence follow the rules for exponents. In economics, the natural logarithms are most often used. Natural logarithms use the base e = 2.71828 , so thatgiven a number e x , its natural logarithm is x . For example, e 3. 6888 isequal to 40, so that the natural logarithm of 40 is 3. 6888. The usual notation for the natural logarithm of x is ln x ;economists and others who have forgotten that logarithms to the base 10 also exist sometimes write log x . Rules for operations are very similiar to those for exponents. ln ab = ln a + ln b ln a/b = ln a - ln b ln a b = b ln aThere is an economically very useful approximate relationship: ln x2 - ln x 1 = PERCENT CHANGE in xThe importance of the natural logarithms in economics comes from the fact that x = e r t will give the value of the variable x at time t if it is continuously compounded at growth rate r We can therefore calculate the present value of a sum S to be received t years in the future as S / e r t = Se -rt since the negative exponent will indicate division. Back to HOME page