Mathematical headaches? Problem solved!
Hi, I'm Colin, and I'm here to help you make sense of maths
Mathematical headaches? Problem solved!
Hi, I'm Colin, and I'm here to help you make sense of maths
I love a good logarithm. Logarithms are a standby for when I want to work something out ninja-style, and there’s something very satisfying about taking something horrible in the powers, bringing it down to the working line, and finding that it wasn’t so horrible after all. I’m an old hand,
Read More →Via @markritchings, an excellent logs problem: If $a = \log_{14}(7)$ and $b = \log_{14}(5)$, find $\log_{35}(28)$ in terms of $a$ and $b$. One of the reasons I like this puzzle is that I did it a somewhat brutal way, and once I had the answer, a much neater way jumped
Read More →“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 →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 →"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 →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 →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 →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 →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 →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
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