The Moderators have been talking about methods to approximate logarithms given only a little bit of information; namely, \(\log_{10}2\approx0.301\) and \(\log_{10}3\approx0.477.\) I'm going to go through a couple methods to provide estimates.
It should be noted that the true value of \(\log_{10}7\) is \(0.845098...\)
Let's start off on a large scale. Obviously, \(\log_{10}6<\log_{10}7<\log_{10}8.\) To find \(\log_{10}6\) and \(\log_{10}8,\) we use the given information. \(\log_{10}6=\log_{10}2+\log_{10}3\approx0.301+0.477=0.778.\) Also, \(\log_{10}8=3\log_{10}2\approx3\times0.301=0.903.\) So \(0.778<\log_{10}7<0.903.\) Not a very good estimate. Even if we take the average of these, we get \(0.8405,\) which is only accurate to two decimal places.
Let's be a little more creative in our thinking. The set of numbers \(\{48,49,50\}\) includes numbers whose logarithms can be expressed with our given information and a power of \(7.\) Once again, \(\log_{10}48<\log_{10}49\log_{10}50\Rightarrow\) \(\log_{10}48<2\log_{10}7<\log_{10}50.\) The only difference here is that \(50\) has a power of \(5\) in its factorization, but \(\log_{10}5\) can easily be eliminated by noticing that \(\log_{10}50=\log_{10}\frac{100}{2}=\log_{10}100-\log_{10}2=2-\log_{10}2.\)
\(\log_{10}48=4\log_{10}2+\log_{10}3\approx1.681.\) Also, \(\log_{10}50=2-\log_{10}2\approx1.699.\) Dividing by \(2,\) we find that \(0.8405<\log_{10}7<0.8495.\) This is better. Not to mention, taking the average of these values yields \(\log_{10}7\approx0.8450,\) accurate to \(4\) decimal places.
What method would you use to calculate \(\log_{10}7?\) Please share what you think!
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Top Newest\[\begin{eqnarray} 2400 & \approx & 2401 \\ \log 2400 & \approx & \log 2401 \\ \log (2^3 \times 3 \times 10^2) & \approx & \log 7^4 \\ \log 2^3 + \log 3 + \log 10^2 & \approx & 4 \log 7 \\ 3 \log 2 + \log 3 + 2 & \approx & 4 \log 7 \\ \log7 & \approx & 0.8450 \\ \end{eqnarray} \] – Pi Han Goh · 2 years, 5 months ago
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So, that begs the question of
1. How do I make such an amazing observation that \( 2400 \approx 2401 \)?
2. How do I know what the next approximation should be?
Think about this some, before reading my other comment. – Calvin Lin Staff · 2 years, 5 months ago
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I don't have a solid answer, but I'm thinking we should find larger pairs of numbers such that their percentage difference is as small as the previous, like: \(6^5 \approx 7777 = \frac {7}{9}(10^4 - 1) \approx \frac {7}{9} \times 10^4 \) or \(6^7 \approx 280000 = 10^4 \times 2^2 \times 7 \), but both methods yields \(0.844\) as the approximation, so I'm kinda stumped right now.
In the meantime, lemme peruse your other comment. =) – Pi Han Goh · 2 years, 5 months ago
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There's a 0.04% error in approximating 2401 as 2400, but works out quite well :) +1 – Azhaghu Roopesh M · 2 years, 5 months ago
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@Calvin Lin and @Daniel Liu were the moderators discussing this in our messageboard. – Trevor B. · 2 years, 5 months ago
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Cool!! So you guys have a message board of your own? – Azhaghu Roopesh M · 2 years, 5 months ago
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The general idea is that if you can bound \( a < 7 ^ n < b \), then you should also look at how \( ab \) compares to \( 7^{2n} \). This is akin to the Root approximation - bisection method.
For example, from above, we have \( 6 < 7 < 8 \), which tells us we should compare \( 48 \) with \( 49 \), and work with \( 48 < 49 < 8^2 \).
Explicitly, we can use the following series of inequalities:
Step 1: \( 6 < 7 < 8 \). Comparing \( 6 \times 8 \) with \( 7^2 \), we have \( 48 < 7^2 \). Hence, this leads to:
Step 2: \( 48 < 7^2 < 8^2 = 64\). Comparing \( 48 \times 64 \) with \( 7^4 \), we have \( 7^4 < 3072 \). Hence, this leads to:
Step 3: \( 2304 = 48^2 < 7^4 < 3072 \). Comparing \( 2304 \times 3072 \) with \( 7 ^ 8 \), we have \( 7^ 8 < 7077888 \). Hence this leads to:
Step 4: \( 5308416 = 2304^2 < 7^8 < 7077888 \).
We can continue this process indefinitely to bound \( \log 7 \) as tightly as we want to. At step 1, we have \( 0.778 < \log 7 < 0.903 \), and at step 4 we have \( 0. 840 < \log 7 < 0.856 \).
Question: Why did I say that this is "akin to the Root approximation - bisection method"?
(see comments below for the explanation)
Note: As it turns out, we could make the observation that 50 a much better bound. With \( 48 < 49 < 50 \), and so we should compare \( 48 \times 50 \) with \( 49^2 \). – Calvin Lin Staff · 2 years, 5 months ago
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Because you want to find values \(a, b\) such that the inequality is satisfied \(a < 7^n < b \)
Taking log to both sides \( \log a < n \log 7 < \log b \)
Bisection method implies we must minimize the difference between \( \frac { \log a + \log b }{2} \) and \( n \log 7 \).
Set both of them to be equal implies we want to compare \( ab \) to \( 7^{2n} \) – Pi Han Goh · 2 years, 5 months ago
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Yes. We are performing the bisection method on \( \log a < n \log 7 < \log b \). The midpoint of this interval is \( \frac{ \log a + \log b } { 2 } = \log \sqrt{ ab } \). We then compare this to \( n \log 7 \), which is in essence comparing \( ab \) to \( 7^{2n} \).
In this way, we don't have to think about what the next number is, or figure out a fancy combination to bound it tightly. We just have to do the bisection method many many times (or automate it with a program), to get the interval down to as small as we want it to. – Calvin Lin Staff · 2 years, 5 months ago
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"It is independent of the number of sig figs / decimal places that we started off with."
I was under the impression, you can only find up to the number of SF given. But this just blew my mind. It's like saying:
"Hey, my calculator is kinda bad, it only let shows that \( \log 2 = 0.3 \) and \( \log 13 = 1.1 \) to one decimal place, can you help me find the value of \( \log 11 \) to a gazillion decimal places?"
Thanks, I learned something new today!! – Pi Han Goh · 2 years, 5 months ago
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Indeed. See my conversation with Kenny Lau below. It is a common misconception (likely from Physics error analysis) that your calculations cannot be more accurate than the degree of significance that you started out with.
For example, even if your calculator can only multiply whole numbers, and you want to approximate \( \sqrt{2} \), you can do much better than \( 1 < \sqrt{2} < 2 \). For example, by showing that \( 1414^2 < 2000000 < 1415^2 \), we can conclude that \( 1.414 < \sqrt{2} < 1.415 \). We can get \( \sqrt{2} \) to any degree of accuracy (with sufficient patience / computing power). – Calvin Lin Staff · 2 years, 5 months ago
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543543
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Well, any method based on the two given information can only be correct to 3 significant figures (or decimal places), because the information itself is correct to only 3 significant figures. – Kenny Lau · 2 years, 5 months ago
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Not necessarily. For example, if we know \( \log 7 ^ {10} \) to 3 decimal places, then we can get \( \log 7 \) to 4 decimal places. – Calvin Lin Staff · 2 years, 5 months ago
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It would not be accurate to 3 decimal places. Log 7^10 would only be accurate to 3 significant figures which is 2 decimal places. – Kenny Lau · 2 years, 5 months ago
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I've added a way to show how we can get an arbitrarily tight bound for \( \log 7 \). It is independent of the number of sig figs / decimal places that we started off with. – Calvin Lin Staff · 2 years, 5 months ago
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