Finding an intuitive understanding of $\operatorname e$ and natural logs...

Question

http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

"this is wild! $e^x$ can mean two things:

x is the number of times we multiply a growth rate: 100% growth for 3 years is $e^3$ x is the growth rate itself: 300% growth for one year is $e^3$. Won’t this overlap confuse things? Will our formulas break and the world come to an end?

It all works out. When we write:

the variable x is a combination of rate and time.

Let me explain. When dealing with continuous compound growth, 10 years of 3% growth has the same overall impact as 1 year of 30% growth (and no growth afterward).

10 years of 3% growth means 30 changes of 1%. These changes happen over 10 years, so you are growing continuously at 3% per year. 1 period of 30% growth means 30 changes of 1%, but happening in a single year. So you grow for 30% a year and stop."

I don't understand why 10 year of 3% growth means 30 changes of 1%. I don't understand why they are making this comparison here? I read this site but this is the part that i'm kind of hung up on?

Answer 1

Let's back up a bit. Recall how compound interest works when there are $n$ compounding periods per year:

$$ A=P\left(1+{r\over n}\right)^{nt}. $$

Now thinking of there being more and more and more compounding periods in a year (every month, every day, every hour, every minute, every second, etc.) so that we are taking the limit as $n\to\infty$ in the above expression. This is called continuously compounded interest and is related to a base $e$ exponential function via $$ A=\lim_{n\to\infty}P\left(1+{r\over n}\right)^{nt}=P\lim_{n\to\infty}\left[\left(1+{r\over n}\right)^{n}\right]^t=P[e^r]^t=Pe^{rt}, $$ where we have used one of the definitions of $e$: $$ e:=\lim_{n\to\infty}\left(1+{1\over n}\right)^n \implies e^r=\lim_{n\to\infty}\left(1+{r\over n}\right)^n \implies e^{rt}=\lim_{n\to\infty}\left(1+{r\over n}\right)^{nt}. $$

So a key thing you're missing in your analysis is non-continuous compounding vs. continuous compounding which moves us from a base $1+{r\over n}$ exponential function to a base $e$ exponential function.

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So take your example: $100\%$ growth ($r=1$), continuously compounded for $3$ years results in $A=Pe^{1\cdot 3}=Pe^3$, i.e., the new amount is $e^3$ times the starting principal $P$.

On the other hand, $10$ years of $3\%$ growth ($r=0.03$) compounded continuously results in $A=Pe^{0.03\cdot 10}=Pe^{0.3}$.

These two are not the same!

Answer 2

Referring to your question, ""10 years of 3% growth means 30 changes of 1% ""

what is meant by above sentence is - final value = (e^3%)^10 = (e^1%)^30 = e^0.3 Clearly they are the same because they produce the same output.

""1 period of 30% growth means 30 changes of 1%, but happening in a single year. So you grow for 30% a year and stop""

what is meant by above sentence is - final value = (e^30%)^1 = e^0.3. Hence they are all the same in terms of producing the same final value i.e. e^0.3