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  Home : Math & Science : Math Study Guides : Algebra II : Logarithmic Functions : Two Special Logarithmic Functions
Logarithmic Functions
  
 
Two Special Logarithmic Functions
The Common Logarithmic Function
The function f (x) = log10x 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).
Figure 2.1: f (x) = log x
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.


e1 2.71828  
e2 7.38906  
e3 20.0855  
e4 54.5982  
     

The function f (x) = logex 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 ex = x and eln x = x.

Examples:
lne4 = 4.
ln = - 2.
ln62 4.1271.
ln230 5.4381.
ln0.04 - 3.2189.
Figure 2.2: f (x) = ln x
Compound Interest and the Natural Logarithm
One of the uses of e is in computing compound interest, using the equation
A = Pert, 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 = Pert = 600e(0.086)(4) = 600e0.344 600(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 = Pert
1822.12 = 1000er(6)
1.82212 = e6r
ln 1.82212 = ln e6r
0.6 = 6r
r = 0.1

The interest rate of the account is 10%.
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