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.