Logarithmic Functions

Two Special Logarithmic Functions

The Common Logarithmic Function

The function f (x) = log₁₀ x is called the common logarithmic function. The common log function is often written as f (x) = log x -- when log is written without a base, the base is assumed to be 10. The " log " button on most calculators means log₁₀ .

Examples:
log1000 = 3 .
log0.01 = - 2 .
log45 1.653 (using a calculator).

The Natural Logarithmic Function

The number 2.71828… occurs often in mathematics and nature. For this reason, it is given a special name: e . Like Π , e is a number that mathematicians use often, and it is an irrational number, so it does not repeat or terminate. e has been calculated to many decimal places, but it is often rounded to e 2.718281828 .

The function f (x) = log_e x is called the natural logarithmic function. The natural log function is often abbreviated f (x) = ln x --this is the way it appears on most calculators. ln e ^x = x and e ^ln x = x .

Examples:
lne ⁴ = 4 .
ln = - 2 .
ln62 4.1271 .
ln230 5.4381 .
ln0.04 - 3.2189 .

Compound Interest and the Natural Logarithm

One of the uses of e is in computing compound interest, using the equation

A = Pe ^rt , where:

P = the amount of money in the original account,
r = the yearly interest rate,
t = the number of years the money is in the account, and
A = the amount of money in the account after t years.

Example 1: If $600 is put into an account which yields a yearly compound interest rate of 8.6%, how much money is in the account at the end of 4 years?

A = Pe ^rt = 600e ^(0.086)(4) = 600e ^0.344 600(1.411) = $846.35
The account has $846.35 at the end of 4 years.

Example 2: If $1000 is put into an account for 6 years, it yields $1822.12. What is the yearly compound interest rate of the account?

A = Pe ^rt
1822.12 = 1000e ^r(6)
1.82212 = e ^6r
ln 1.82212 = ln e ^6r
0.6 = 6r
r = 0.1

The interest rate of the account is 10%.