x -a = 1 / xa
Negative exponents give the reciprocal of the positive expontne For example
x2 times x3 = x 5
Dividing variables raised to a power involves subtracting their exponents.
x5 / x3 = x 2
Exponentiation of variables raised to a power involves multiplying the exponents.
x5 squared = x 10
Note: There are no easy rules for addition and subtraction of variables raised to a power.
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 .
There is an economically very useful approximate relationship:
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
since the negative exponent will indicate division.