Browsing category logarithms

The Mathematical Ninja and Logs Base 2

The student’s shoulder twitched slightly as he said “So I need to work out $\log_2(10)$…” and the crash of the cane against the table reminded him that the calculator was off-limits. “I think you can estimate that yourself,” said the Mathematical Ninja. “Uh… ok. There’s a change of base formula,

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A few logarithmic tricks

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,

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A logs puzzle

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

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Powers

“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}

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Ask Uncle Colin: An Uncommon Logarithm

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

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How the Mathematical Ninja approximates $\ln(5)$

"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

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Ask Uncle Colin: How do I find the power?

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

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Ask Uncle Colin: A Huge Power Of Two

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

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Ask Uncle Colin: How did they get $\ln(50)$?

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 $-\ln(0.02)=\ln(50)$,

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A frankly ludicrous bit of paper arithmetic

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

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