The Rate of Growth of Continuous Processes A natural logarithm, also known as a common logarithm, is the logarithm of a number that has been raised to some positive power. It is denoted as ln (or sometimes N) and canbe calculated using the following formula: ln(x) = x / (1 + x) The base of the natural logarithm is e.Euler's constant is a mathematical constant, usually denoted by the lowercase Greek letter epsilon ( ε ). It is defined as the base of natural logarithms, with a value ofapproximately 2.71828182845904523536028747135266249775724709369995... The term exponential rate of growth refers to an expression of the rate at which something is growing. There are two different ways you can have an exponential rate ofgrowth: discrete and continuous. A discreet rate is a simple concept. Imagine that I have $1.00 and it's growing at a certain rate, r, and I have a certain number of time intervals, t. In this situation, r representshow much my money would grow in discreet intervals of time, t. If my rate of return is 100% per year, then r = 1 and t = 1. At the end of one year, I would have $2.00; at the end of two years I would have $4.00; at the end of three years Iwould have $8.00 and so on. But what we are most interested in here is a concept known as continual exponential growth and a special constant known as Euler's constant, also known as e. I'm going toshow you now how to develop an intuition for what this special number e is. Let's suppose that we had an interest rate of 100%. And a clever man said: "If you are willing to pay me 100% for a year, wouldn't you be willing to pay me 50% for sixmonths?" The bank might agree to a 50% interest rate for six months and then it would offer interest on one year's worth of accrued interest at the same rate. This means that if I havea factor of 1 + 1 and I repeat it one time, I will have 2. But let's say that a person has a factor of 1.5 and repeats it twice, thus creating a result of 2.25. And the same clever man might say, well gee, if you agreed to pay meinterest twice a year, wouldn't you agree to pay me four times a year including interest onmy interest? We are calculating 1.25 raised to the 4th power or 1.5 raised to the 2nd power, where this value is our rate per unit time and these are our number of time intervals. So, you'll notice that this is decreasing and this is increasing. If we were paid every three months, we'd receive $2.44 for every dollar. So, a fundamental question to ask is:does this number keep getting bigger forever and ever? When I reduce the time intervals smaller and smaller, does my potential wealth increase and increase to infinity? Surprisingly perhaps, the answer is “no.” It doesincrease, but eventually it levels off. Let's examine what that looks like in a graph. If I am receiving interest every month, my factor would be 1.08. It would repeat 12 times, so I would have 1.08 to the 12th power, or 2.613. If I am getting paid interest everyweek, this amount would be multiplied by 1.019 raised to the 52nd power, which would be2.693. If I was paid interest every day, this would be 1.002739 times 365. That would equal2.7146.
You will notice that the numbers are not increasing as rapidly and there are 8,760 hours in a year. If I were to raise, I would be entitled to $2.71813. At the end of 525,600minutes in a year, I would receive $2.71828. For every 31,536,000 seconds in a year, Iwould receive $2.71828. Although this number continues on to five significant digits, it has stabilized as I get to a minute or second. The number e, approximately 2.71828, is known as Euler's constant. Let's consider an example of a problem. Let's consider a baby elephant that grows continuously at aknown rate. Let us assume that I have an elephant that weighs 200 kilograms. It grows at a continuously compounded annual rate of 5%, so its growth is continuous and compoundedannually at 5%. I would like to know what this elephant will weigh in three years. It willweigh (200 kg)*e to the (.05)t, which is 3. It would weigh 200 times e to the 0.15, or 232.4kg. In addition to e, we have the logarithm of x to the base e, written ln(x), which is called the natural logarithm. The natural log is used to calculate natural growth rates because it reflects a continuous rate of growth. Let us assume that we have some rabbits with unlimited food. These animals increase in mass at a rate of 200% per year. Suppose that we start with a population ofmale and female rabbits that weighs 10 kg. They have unlimited food and resources, andwhat we want to know is: If their growth rate is 200% per year, how many years will it takefor them to weigh as much as the Earth, which is 5.972 x 10 to the 24 kg? The way we set up this problem is that we have 5.972 x 10 to the 24 kg = 10 kg times e to the 2t, where 2 is equal to 200%, and t is equal to the number of years required tocreate a 5.972 x 10 to the 24 kg worth of rabbits, okay? We can divide both sides of theequation by 10. This gives us 5.972 x 10 to the 23 = e to the 2t, okay? We'll use a littletrick now, which is that we're going to take logarithms of both sides of this equation. Thus,we have log(5.972 x 10 to the 23rd) = 2t, so that t is equal to (the natural log of thisnumber)/2. You can use your pocket calculator, or Excel, or your computer to find thenatural log of this number and divide by 2. This is a useful concept to know when trying to determine your projected rate of return on investment. This is a result of exponential growth and continuous compounding,which are related by logarithms.