Browsing category logarithms

Ask Uncle Colin: A Limiting Issue

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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How the Mathematical Ninja estimates logarithms

"$\ln$", 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

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Ask Uncle Colin: A logarithmic coincidence?

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

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The Mathematical Ninja and the Powers of 10

"So that works out to be $10^{1.6}$," said the student, reaching for the calculator -- and, of course, recoiling as the Mathematical Ninja yelled "yeeha!" and lasso-ed it out of her hand. "Forty," he said. "Too high by half a percent or so. 39.8." The student paused. She would normally

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Ask Uncle Colin: Are the log laws… lacking?

Dear Uncle Colin, I have an equation to solve: $\ln(x^2) = 2 \ln(4)\, x \ne 0$. I tried to solve it by applying the log laws: $2 \ln(x) = 2 \ln(4)$, so $x=4$. However, a bit of thought shows that $x=-4$ is also a solution -- but that doesn't seem

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Ask Uncle Colin: a proof with logs in

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can. Dear Uncle Colin, I have a slightly embarrassing problem: I

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Why logs and exponentials undo each other

Someone on the internet asks: I don't get why $\ln(e^x) = e^{\ln(x)}$. Can you explain? Of course I can! Or at least, I can try; the easy answer is to say 'by definition', but that doesn't help you much. $\log_n(x)$ answers the question "what power would I raise $n$ to,

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Why $\log_{10} (2) \simeq 0.3$

“Coincidence?” said the Mathematical Ninja. “I think not!” He looked at his whiteboard pen as if wondering how best to fashion a weapon out of it. He wrote: $10^3 = 1,000$ $2^{10} = 1,024 = 1.024 \times 10^3$ So $10 \ln (2) = 3 \ln(10) + \ln(1.024)$ But $\ln(1.024) \simeq

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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

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Which is larger?

A MathsJam classic question asks: Without using a calculator, which is bigger: $e^\pi$ or $\pi^e$? It's one of those questions that looks perfectly straightforward: you just take logs and then... oh, but is $\pi$ bigger than $e\ln(\pi)$? The Mathematical Ninja says "$\ln(\pi)$ is about 1.2, because $\pi$ is about 20%

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