Finding the Actual Doubling Time (Logarithms and Exponential Models)

Question

This problem involves the Rule of 70, which gives quick estimates of the doubling time of an exponentially growing quantity. If r% is the annual growth rate of the quantity, then the Rule of 70 says

 

Doubling time in years =(aprox) 70/R

Use the Rule of 70 to estimate how long it takes an $8000 investment to double if it grows at the annual rate 2%. Compare with the actual doubling time.

Round your answer for the actual doubling time to three decimal places.


The doubling time estimated with the rule of is: I got this one, 35.

The actual doubling time is Entry field with incorrect answer years.

 

I'm not sure how to go about this.

Answer 1

Your estimate for the doubling time is correct.

 

The actual doubling time can be found by using the Law of Exponential Growth: A = P e^rt.

 

Here, you know that when P = 8000 and r = .02. The problem is to find t when A = 16000.

 

Set up the equation with these numbers plugged in:

 

16000 = 8000 e^.02t

 

Divide both sides by 8000 to isolate the exponential e^.02t:

 

2 = e^.02t .

 

To get at the variable t in the exponent, you must use the natural logarithm, ln. Apply it to both sides:

 

ln 2 = ln (e^.02t) = .02t .

 

There were several things going on in that last step. The logarithm ln was used because the base of the exponential was the number e and this is the corresponding logarithm function to use with that. Also, when I simplified after the first equal sign, I used an important property of ln, namely that ln (e^x) = x for any real number x. This is a REALLY important and useful thing to know about ln and e^x and is the reason why I used ln here.

 

Now, you can solve for t by dividing both sides by .02:

 

t = (ln 2)/.02 ≈ 34.66 years (rounded to two decimal places).

 

This is very close to your estimate. Do you want to know why? I will tell you anyway. The reason is that the decimal for ln 2 is very close to .70. Try it on your calculator and see. This is why the rule is called the rule of 70. When you use the number 70 and the percent (not the decimal), you GET the rule of 70.

 

Surprise!