It equals one.
That’s the short answer. If you typed $\log_e(e)$ into a calculator or just asked a math teacher what is log of e, they’d tell you the answer is 1 and probably move on to the next slide. But honestly? That doesn't tell you anything about why this weird little number matters or why the entire world of finance, physics, and even your morning cup of cooling coffee relies on it.
The number $e$, often called Euler's number, is roughly 2.71828. When you take the logarithm of $e$ with respect to itself, you’re basically asking: "To what power do I have to raise 2.718 to get 2.718?" It’s a self-referential loop. Because any number (except zero) raised to the power of 1 is itself, the answer has to be 1.
But we need to talk about the notation. In most high-level math and science circles, when people say "log of e," they are talking about the natural logarithm, written as $\ln(e)$.
Why does log of e keep popping up?
It’s not just a textbook quirk. The natural log is the language of growth. If you've ever looked at a graph of how a virus spreads or how interest compounds in a high-yield savings account, you're looking at $e$ in the wild.
Think about it this way. Most of us learn "base 10" math because we have ten fingers. It feels natural to us. But nature doesn't have fingers. Nature grows continuously. Trees don't wait until the end of the year to add a ring; they are constantly, microscopic-bit by microscopic-bit, expanding. That continuous growth is what $e$ represents.
Leonhard Euler, the Swiss genius who popularized the constant, wasn't just playing with symbols. He realized that there is a specific rate of growth where the rate of change is perfectly equal to the value itself. If you have $e$ amount of "stuff," it grows at a rate of $e$. It’s the only number that behaves this elegantly in calculus. When you differentiate $e^x$, you get $e^x$. It’s its own shadow.
The confusion between log and ln
This is where students usually get tripped up. Depending on who you ask, "log" means different things.
If you ask a software engineer or a computer scientist, "log" usually means base 2 because of binary. If you ask a high school student, "log" means base 10. But if you ask a physicist or a mathematician, "log" almost always means the natural log (base $e$).
The expression looks like this:
$$\ln(e) = 1$$
If you accidentally use a common log (base 10), the value of $\log_{10}(e)$ is approximately 0.434. That’s a massive difference if you’re calculating the structural integrity of a bridge or the decay rate of a radioactive isotope. Always check the base. Most modern scientific calculators have a dedicated $\ln$ button just to avoid this mess.
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Real world: From bank accounts to carbon dating
Let’s get away from the abstract symbols for a second. Why should you care?
Consider compound interest. If a bank offered you 100% interest per year, and they compounded it once, you'd have 2x your money. If they compounded it every month, you'd have more. If they compounded it every second—continuously—you wouldn't have infinite money. You’d have exactly $e$ times your original investment.
$\ln(e)$ is the tool we use to "undo" that growth. If $e^x$ tells you how much you'll have after a certain time, the natural log tells you how much time it took to get there.
- Radioactive Decay: Archeologists use the natural log to figure out how old a bone is. They measure how much Carbon-14 is left and use the log of $e$ to work backward to the date of death.
- Rocket Science: The Tsiolkovsky rocket equation uses natural logs to determine how much delta-v a spacecraft has based on its mass ratio. Without understanding what is log of e, we don't get to Mars.
- Cooling: That coffee I mentioned? Newton’s Law of Cooling uses $e$ to predict how fast your latte reaches room temperature.
How to calculate it (if you have to)
You don't really "calculate" $\ln(e)$ because it's a definition. It's like asking how many centimeters are in a centimeter. But if you are dealing with $\ln$ of other numbers, you’re usually looking at a Taylor series expansion or using a lookup table (though nobody uses those anymore).
One interesting property is that the area under the curve of $y = 1/x$ from $1$ to $e$ is exactly 1.
It’s one of those beautiful moments where geometry, algebra, and calculus all collide. You have this curvy, infinite-looking function $1/x$, and yet, if you stop exactly at $e$, the area is a perfect, clean 1.
Common pitfalls to watch out for
Don't assume $\ln(e^x)$ is complicated. It’s just $x$.
The natural log and the exponential function are inverses. They cancel each other out like fire and water. If you see $\ln(e^{500})$, don't reach for a calculator. The answer is 500.
Another weird one? The log of a negative number. In the world of basic real numbers, you can’t take the log of a negative. It’s "undefined." But if you step into the world of complex numbers and use Euler’s Identity ($e^{i\pi} + 1 = 0$), things get wild. You start seeing logs of negative numbers involving $i$ (the imaginary unit). But for 99% of people, just remember: you can't take the log of zero or a negative number.
Is it actually useful for SEO or business?
Surprisingly, yes. Data scientists use logarithmic scales to visualize data that grows too fast to see on a normal graph. If you’re looking at website traffic that goes from 10 visitors to 1,000,000, a linear graph is useless. You’ll just see a flat line and then a vertical spike.
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By using a log scale—specifically one rooted in the natural log—you can see the percentage growth over time. It levels the playing field. It lets you see if your growth rate is accelerating or slowing down, regardless of the raw numbers.
Moving forward with e
Understanding what is log of e is basically your entry point into "grown-up" math. It’s the bridge between the simple counting we do as kids and the complex, continuous world we actually live in.
If you’re a student, stop trying to memorize the decimal places of $e$. It’s 2.718... and it goes on forever without repeating. That’s not the important part. The important part is that $e$ is the "unit" of natural growth. And because it's the unit, its log is 1.
Next time you see a formula with $\ln$ in it, don't panic. Just remember it’s talking about time, growth, and the way things naturally change.
Actionable Steps:
- Check your calculator settings: Ensure you know the difference between the
logbutton (usually base 10) and thelnbutton (base $e$) on your specific device. - Practice the Inverse: Try solving for $x$ in the equation $e^x = 10$. You'll do this by taking the natural log of both sides: $x = \ln(10)$.
- Visualize Growth: If you work in marketing or finance, try switching your chart axis to a "logarithmic scale" in Excel or Google Sheets. It often reveals trends in your data that a standard linear view hides.
- Memorize the Identity: $\ln(e) = 1$ and $\ln(1) = 0$. These two will save you more time in exams and coding than almost any other log property.