#
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.
## Exponents

** Definitions **

**
x**^{a} = x times x times x ... to a total of a times.
x^{ -a } = 1 / x^{a}

Negative exponents give the reciprocal of the positive expontne
For example

**
x**^{ -2 } = 1 / x^{2}

## Operations

** Multiplying ** variables raised to a power involves
** adding ** their exponents.
**
x**^{a} times x^{b} = x^{ a + b }
x^{2} times x^{3} = x^{ 5 }

**
**

** Dividing ** variables raised to a power involves
** subtracting ** their exponents.

**
x**^{a} divided by x^{b} = x^{ a - b }
x^{5} / x^{3} = x^{ 2 }

**
**

** Exponentiation of ** variables raised to a power involves
** multiplying ** the exponents.

**
x**^{a} raised to the b power = x^{ a b }
x^{5} 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 x**_{2} - 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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