After grasping the concept of exponential functions, understanding What Is Natural Logarithm is the next logical step. While math textbooks might present it as the inverse of $e^x$, which can seem somewhat abstract, there’s a more intuitive way to think about it.
The natural log tells you the time needed to reach a certain level of continuous growth.
Imagine you’re investing in something that grows continuously at a rate of 100% per year. To achieve 10x growth, you’d only need to wait approximately $ln(10)$ or 2.302 years. This assumes continuous compounding.
Here’s how e and the natural log relate:
- $e^x$: The amount you have after starting at 1.0 and growing continuously for x units of time.
- $ln(x)$ (Natural Logarithm): The time required to reach amount x, assuming continuous growth from 1.0.
Let’s explore this intuitive explanation further.
Understanding E: The Essence of Growth
The number e represents continuous growth. $e^x$ allows us to combine rate and time. For instance, 3 years at 100% growth is equivalent to 1 year at 300% growth when compounding continuously.
We can convert any rate and time combination (like 50% for 4 years) into a 100% rate for simplicity (resulting in 100% for 2 years). By normalizing the rate to 100%, we can focus solely on the time aspect:
The formula shows the relationship between growth rate, time, and the exponential function.
Therefore, $e^x$ intuitively signifies:
- The amount of growth achieved after x units of time, assuming a continuous growth rate of 100%.
- For example, after 3 time periods, the growth will be $e^3$ = 20.08 times the initial amount.
$e^x$ acts as a scaling factor, illustrating the extent of growth after x units of time.
The Natural Log: Unveiling the Time Factor
The natural log is the inverse function of $e^x$. Its Latin name, logarithmus naturali, gives us the abbreviation ln.
So, what does “inverse” mean?
- $e^x$ takes time as input and produces growth as output.
- $ln(x)$ takes growth as input and produces the time required to achieve it.
Consider these examples:
- $e^3$ equals approximately 20.08. After 3 units of time, the result is 20.08 times the starting amount.
- $ln(20.08)$ is approximately 3. Achieving a growth of 20.08 requires waiting 3 units of time (assuming a continuous growth rate of 100%).
The natural log essentially provides the time necessary to reach a specific growth target.
Navigating the Realm of Logarithmic Arithmetic
Logarithms often appear peculiar. How do they convert multiplication into addition and division into subtraction? Let’s investigate.
What is $ln(1)$? Intuitively, this asks: How long does it take to reach 1x the current amount?
The answer is zero. No time is needed to remain at the current amount!
- $ln(1) = 0$
What about a fractional value? How long to reach 1/2 of the current amount? Knowing that $ln(2)$ is the time to double with 100% continuous growth, reversing the process (using negative time) yields half the current value.
- $ln(.5) = – ln(2) = -.693$
Reversing .693 units of time results in half the current amount. Similarly, $ln(1/3) = – ln(3) = -1.09$. Going back 1.09 units of time provides a third of the current amount.
What about the natural log of a negative number? How long does it take to grow a bacteria colony from 1 to -3?
This is impossible. A negative amount of bacteria is nonsensical.
- $ln(text{negative number}) = text{undefined}$
“Undefined” signifies that no amount of time can produce a negative amount.
The Magic of Logarithmic Multiplication
How long to grow 9x the current amount? While $ln(9)$ works, let’s try a different approach.
Growing 9x can be viewed as tripling (taking $ln(3)$ time) and then tripling again (taking another $ln(3)$ time):
- Time to grow 9x = $ln(9)$ = Time to triple and triple again = $ln(3) + ln(3)$
Any growth number can be decomposed into sequential growth steps. For example, 20x growth can be 2x followed by 10x.
- $ln(a*b) = ln(a) + ln(b)$
The log of a times b equals log(a) + log(b). This holds true when considering the time required for growth. Growing 30x can be achieved by waiting $ln(30)$ or by waiting $ln(3)$ to triple, then waiting $ln(10)$ to grow 10x again.
What about division? $ln(5/3)$ asks: How long does it take to grow 5 times and then take 1/3 of that?
Growing 5 times takes $ln(5)$. Growing 1/3 takes $-ln(3)$ time. So:
- $ln(5/3) = ln(5) – ln(3)$
Growing 5 times and then going back in time to have a third of that amount leaves 5/3 growth.
- $ln(a/b) = ln(a) – ln(b)$
Multiplication of growth becomes addition of time, and division of growth becomes subtraction of time. It’s more important to understand these rules, rather than just memorize them.
Applying Natural Logs to Various Growth Rates
What if the growth rate isn’t the convenient 100%?
The “time” derived from $ln()$ incorporates both rate and time (“x” in $e^x$). A 100% rate simplifies the calculation, but other rates are usable.
To achieve 30x growth: $ln(30)$ equals approximately 3.4. This means:
- $e^x = text{growth}$
- $e^{3.4} = 30$
Intuitively, this means “100% return for 3.4 years results in 30x growth.”
The formula shows the calculation of achieving 30x growth over 3.4 years with a 100% return rate.
Modifying “rate” and “time” is possible as long as their product remains 3.4. To get 30x growth at a 5% return:
- $ln(30) = 3.4$
- $text{rate} * text{time} = 3.4$
- $.05 * text{time} = 3.4$
- $text{time} = 3.4 / .05 = 68 text{years}$
$ln(30) = 3.4$ means 3.4 years are needed at 100% growth. Doubling the growth rate halves the time required.
- 100% for 3.4 years = 1.0 * 3.4 = 3.4
- 200% for 1.7 years = 2.0 * 1.7 = 3.4
- 50% for 6.8 years = 0.5 * 6.8 = 3.4
- 5% for 68 years = .05 * 68 = 3.4
The natural log is compatible with any interest rate or time, as long as their product is constant.
Example: The Rule of 72 Explained
The Rule of 72 is a shortcut to estimate the time required to double your money. Let’s derive it and understand its intuition.
How long does it take to double your money at 100% interest, compounded annually?
Using natural logs for continuous rates with yearly interest introduces a slight inaccuracy. However, at reasonable interest rates (e.g., 5%, 6%, or 15%), the difference between yearly compounded and fully continuous interest is negligible. Therefore, the formula provides a reasonable estimate.
The question then becomes: How long to double at 100% interest? ln(2) = .693. It takes .693 units of time to double your money with continuous compounding at 100%.
With a different interest rate, rate * time = .693. Therefore:
- rate * time = .693
- time = .693/rate
At 10% growth, it would take .693 / .10 or 6.93 years to double.
Multiplying by 100 simplifies the calculation:
- time to double = 69.3/rate (where rate is in percent).
Doubling time at 5% growth is approximately 69.3/5 or 13.86 years. Approximating 69.3 with 72 (highly divisible) gives:
- time to double = 72/rate
This is the Rule of 72!
Finding the time to triple uses ln(3) ~ 109.8:
- time to triple = 110 / rate
The Rule of 72 applies to interest rates, population growth, bacteria cultures, and other exponentially growing phenomena.
Conclusion
The natural log represents the time required for exponential growth. Considering e as the universal rate of growth makes ln the universal way to determine growth time.
When encountering $ln(x)$, interpret it as “the time to grow to x”.
Appendix: The Natural Log of E
What’s $ln(e)$?
- The mathematical answer: Because they are defined as inverse functions, $ln(e) = 1$
- The intuitive answer: ln(e) is the time needed to reach “e” units of growth (about 2.718). But e is the amount of growth after 1 unit of time, so $ln(e) = 1$.
Think intuitively.
