# Powers of $e$ revisited: Secrets of the Mathematical Ninja

The Mathematical Ninja woke up at 8:58, and opened his other eye.

“$e^{12}$?” asked his alarm clock.

“$160,000$,” said the Mathematical Ninja, and sat bolt upright. He leapt out of his sleeping corner, somersaulted across the room, landing in front of the whiteboard just as the student arrived.

“$e^{12}$?” he asked the student by way of greeting.

“Uh… good morning?” said the student. “Well, $3^{12}$ is $729^2$, so that’s about 500,000…”

“531,441,” said the Ninja.

“But that’s an over-estimate, so I could multiply by $0.9^{12}$, which is about $e^{-1.2}$, a bit less than a third? 150,000 or so?”

The Mathematical Ninja nodded. “$2.7^{12}$ is 150,094 and a bit,” he agreed. “I suppose you *could* do it that way.”

This, the student supposed, was as close to praise as the Mathematical Ninja was likely to get. “How would you do it?”

“Oh, easy!” said the Mathematical Ninja. “$\ln(10)$ is about 2.3, or pretty close to $\frac{30}{13}$.”

“Obviously,” said the student.

“So, $e^{12}$ is about $10^{12 \times \frac{13}{30}}$.”

“144… 156 over 30 is five and a bit?”

“Well, $\frac{13}{30}$ is a third plus a tenth, so it’s 4 plus 1.2 or 5.2”

“OK, so we’re at $10^{5.2}$. More than 100,000, but less than 200,000, like I said, because $\log_{10}(0.2)$ is between 1 and 2.”

“It’s about 1.6,” said the Mathematical Ninja. “So, it’s about 160,000. What’s $e^{40}$?”

“Good God, I’ll be lucky to be in the right ballpark with that one. A third is 13.3, a tenth is 4, so 17.3… About $2 \times 10^{17}$?”

The Mathematical Ninja nodded. “Good to one significant figure. You can adjust slightly - the 13/30 figure is off by a factor of $\frac{1}{450}$ or so - so by the time you get to 40, your final answer is off by nearly 10%. The $10^{17}$ is great, but we need to deal with the $10^{\frac 13}$, which is 2.15 or so. Adding on the 10% is 2.35 ish, so we get $2.35 \times 10^{17}$.”

“May I?” asked the student, pointing at a calculator.

The Mathematical Ninja smirked. “Go ahead,” he said. “Make my day.”

* Thanks to Pi Guy for reminding the Mathematical Ninja to revise his methods (his version is here.)

* Edited 2014/08/11 to fix a LaTeX error and a dialogue mistake.