Rules for Exponents

In working with production functions and growth models, we often have to work with exponents, including fractional exponents. A brief review of the basics follows.



xa = x times x times x ... to a total of a times.

x -a = 1 / xa

Negative exponents give the reciprocal of the positive expontne For example

x -2 = 1 / x2


Multiplying variables raised to a power involves adding their exponents.

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.

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 that given a number e x , its natural logarithm is x . For example, e 3. 6888 is equal 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 a

There is an economically very useful approximate relationship:

ln x2 - ln x 1 = PERCENT CHANGE in x

The 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.

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