Posted in logarithms, ninja maths.

“Here’s a quick one,” suggested a fellow tutor. “Prove that $2^{50} < 3^{33}$.” Easy, I thought: but I knew better than to say it aloud. First approach “I know that $9 > 8$,” I said, checking on my fingers. “So if $2^3 < 3^2$, then $2^{150} < 3^{100}$ and $2^{50}

Read More →
Posted in ask uncle colin, logarithms.

Dear Uncle Colin, I'm a bit stumped by a logs question with a variable base: $\log_{\sqrt[3]{x+3}}(x^3 + 10x^2 + 31x + 30) = 9$. I know the basics of logarithms, but this is currently beyond me. -- Obtaining Underwhelming Grade, Having To Review Every Definition Hello, OUGHTRED, and thanks for

Read More →
Posted in logarithms, ninja maths.

"Isn't it somewhere around $\phi$?" asked the student, brightly. "That number sure crops up in a lot of places!" The Mathematical Ninja's eyes narrowed. "Like shells! And body proportions! And arrawk!" Hands dusted. The Mathematical Ninja stood back. "The Vitruvian student!" The student arrawked again as the circular machine he

Read More →
Posted in ask uncle colin, logarithms.

Dear Uncle Colin, The equation $67.5 = 10(1.0915)^{10-n} + 30(1.0915)^{10-2n}$ cropped up in a question. Excel can solve that numerically, but I can't solve it on paper! Any ideas? Problems Occur When Exponentials Recur Hi, POWER, and thanks for your message! That's an ugly one. First thing: beware of the

Read More →
Posted in ask uncle colin, logarithms, powers.

Dear Uncle Colin, I've been asked to find $2^{64}$ without a calculator, to four significant figures. How would you go about this? -- Large Exponent, Horrific Multiplication, Extremely Repetitive Hi, LEHMER! To get a rough answer, I'd usually start with the rule of thumb that $2^{10} \approx 10^3$. I'd conclude

Read More →
Posted in ask uncle colin, logarithms.

Dear Uncle Colin, I get $-\frac{\ln(0.02)}{0.03}$ as my answer to a question. They have $\frac{100\ln(50)}{3}$. Numerically, they seem to be the same, but they look completely different. What gives? -- Polishing Off Weird Exponents, Really Stuck Dear POWERS, What you need here are the log laws (to show that $-\ln(0.02)=\ln(50)$,

Read More →
Posted in arithmetic, logarithms.

Some time ago, someone asked Uncle Colin what the last two digits of $19^{1000}$ were. That caused few problems. However, Mark came up with a follow-up question: how would you estimate $19^{1000}$? I like this question, and set myself some rules: No calculators (obviously) Only rough memorised numbers ($e \approx

Read More →
Posted in ask uncle colin, logarithms.

Dear Uncle Colin, I have a problem with a limit! I need to figure out what $\left( \tan \left(x\right) \right)^x$ is as $x \rightarrow 0$. -- Brilliant Explanation Required Now! Our Understanding's Limited; L'Hôpital's Inept Right, BERNOULLI, stop badmouthing L'Hôpital and let's figure out this limit. It's clearly an indeterminate

Read More →
Posted in logarithms, ninja maths.

"$\ln$", said the student, "of 123,456,789." He sighed, contemplated reaching for a calculator, and thought better of it. "18.4," said the Mathematical Ninja, absent-mindedly. "A bit more. 18.63." The student diligently wrote the number down, the Mathematical Ninja half-heartedly pretended to visit some violence on him, and the student squeaked

Read More →
Posted in ask uncle colin, binomial, logarithms.

Dear Uncle Colin, I noticed that $2^{\frac{1}{1,000,000}} = 1.000 000 693 147 2$ or so, pretty much exactly $\left(1 + \frac{1}{1,000,000} \ln(2)\right)$. Is that a coincidence? Nice Interesting Numbers; Jarring Acronym Dear NINJA, The easiest way to see that it's not a coincidence is to check out $3^{\frac{1}{1,000,000}} $, which

Read More →