The function f (x) = log_{10} 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 log_{10} .
Examples:
log1000 = 3
.
log0.01 = - 2
.
log45 1.653
(using a calculator).
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 .
e ^{1} | 2.71828 | ||
e ^{2} | 7.38906 | ||
e ^{3} | 20.0855 | ||
e ^{4} | 54.5982 | ||
… |
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
.
One of the uses of e is in computing compound interest, using the equation
A = Pe ^{rt} , where:
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%.