You may remember from math class that the number 2 raised to the power of three is represented as 23, which is equivalent to 2 times 2 times 2, or 8.
Consider the number 10, raised to the power of 2. This is equivalent to 10 times 10, or 100. We could say that the logarithm of 100 is the power to which the number 10 is raised in order to reach 100: in this case, it is 2. We're using a logarithm to the base 10; that is to say, the logarithm of 100 to the base of 10 is 2, because 10 raised to the second power equals 100.
In mathematical form, our example looks like this:
Log10(102) = 2
For financial applications, we generally use natural logarithms, which are logarithms to the base e, a constant number that is approximately 2.718. This number, like pi, recurs often in mathematicians' work.
Instead of writing loge, we generally use the notation ln. And we can say that ln e1 = 1, and ln e2 = 2, and so on.
A Practical Application
A common example of natural logarithms in finance is calculating how long it will take a bank deposit at a set interest rate to reach a specified higher amount. The equation for compound interest is:
A is equal to the final amount of money in the account after t, the number of years the account is growing in value. The annual interest rate is represented by r, and the original amount of money in the account is shown with P.
Let's take an example of a saver who puts $2,000 into an account at a yearly compounded interest rate of 4.1% and wants to know how long it will take before the account grows to $4,000.
Solving this equation becomes simple with natural logs.
A = Pert
4,000 = 2,000e0.041(t)
2 = e0.041(t)
ln 2 = ln e0.041(t)
Now, consulting a natural log table (you can find one at sosmath.com), or using a natural log function on a calculator, we learn that ln 2 = 0.301030.30103 = 0.041(t)
t = 7.3
So it would take 7.3 years for the investment to double.