If 2^n and 3^10 have the same number of digits, then n could be…
Vložit
- čas přidán 8. 09. 2024
- This is a mock college entrance exam problem for high school students in Taiwan. The original post did it the elementary school way but is there a better way?
This question is from Threads: www.threads.ne...
Shop my math t-shirts & hoodies on Amazon: 👉 amzn.to/3qBeuw6
-----------------------------
I help students master the basics of math. You can show your support and help me create even better content by becoming a patron on Patreon 👉 / blackpenredpen . Every bit of support means the world to me and motivates me to keep bringing you the best math lessons! Thank you!
-----------------------------
#math #algebra #mathbasics
I am begging you, please explain like I’m five. Reddit precalculus r/Homeworkhelp
czcams.com/video/F-gpF2gL97Y/video.html
For computing log(3), one could also reason that since 3 is just a bit smaller than √10 , log(3) would be just a bit smaller than 0.5
3¹⁰=(3²)⁵=9⁵
@@KrasBadan You're missing out steps.
9^5 =9*9^4>1.5^4*9^4=13.5^4>10^4
So it cannot be 4 digits or less
In Croatia we were always allowed to use the logarithmic tables (which usually came also with trigonometry tables along), while not being able to use calculators. Worked out pretty good.
Honestly, you can just compute the exponentiation manually. It's not that big of a number. 3^10=9^5=729*81, which is larger than 700*80=56000 and smaller than 729*100=72900. So, we are asked for 2^n with exactly 5 digits. This should be answerable instantly with 2^14=16384, 2^15=32768, and 2^16=65536
I am surprised to know that Taiwan students have to memorize log 2 and log 3. In Hong Kong, we are allowed to use calculator. I will probably use the Newton's method to calculate log 3 if calculator is not allowed (although I still need to know ln 10 for Newton's method to work)
@@cyrusyeung8096 actually, some people told me that those numbers were given on the exam.
@@bprpmathbasics I see. That is much better for students.
I had a math teacher who was from Taiwan and he did tell me lg2 and lg3 must be memorised. You can then use these two to figure out lg of all the other numbers under 10. (Except for lg7)
IDK how is it today but back in the 2000's when I was in High School, we were given log table on the back or at the very end of the question paper as references. I'm from Indonesia btw.
@@Ninja20704 doesn't this also exclude log 5 or am I missing something?
The problem is log 2 and 3. At least in my high school era these are not memorized.
Did they not allow log tables?
@@pentasquare No. And the examinations do not require them.
I don't have the log values memorized, but I do know a few powers of 2. Estimating 3^10 via 9^2 x 9^2 x 9 as 81 x 81 ~= 6400 x 9 --> 5 digits, I started at 2^10 ~= 1000 then doubled up a few times (2000, 4000, 8000, 16000 ~= 2^14, ..) to get to 14, 15, and 16 as possible candidates
You could say it's 5 digits with just 3^10 = 9^5
9^5 is obviously less than 10^5 which is the smallest 6 digit number and bigger than 50,000 (simply thinking about the first digit when multiplying by 9 tells you that, 9×9>80; 80×9>700; 700×9>6,000; 6,000×9>54,000).
So 3^10 is somewhere between 50,000 and 99,999; either way it's definitely 5 digits.
You should tell us more about those Taiwanese special log digits
BlackpenRedpenWhitePaper ??
Haha yes
I found a pattern that when 3^2n then the number of digits is n and for 3^(2n-1) it's also n similarly for 2^3n the number of digits is n for 2^(3n-1) the digits number is n too and for 2^(3n-2) number of digits is n-1
Darn interesting problem with an even more interesting solution
I don't know log10(3) but I do know that 10 to the 0.5 is between 3 and 4, so it's probably 0.4(something)
and I can recite 2's powers up to 2097152, so that's not a problem.
I remembered log2 as 0.3 after 50 years of not needing it. And I remembered that 3 is nearly in the center of my old slide rule C and D scales so log 3 is nearly 0.5
I saw to use logarithm solution right away .
Some of us are old enough to remember having a book of log tables which we were expected to use in school - we weren't expected to remember any log values!!
Here's a slightly different perspective on the Problem:
Q. How many binary digits, n, are required to represent the trinary number 3^10 ?
i.e. 2^n = 3^10
The solution requires careful interpretation of the problem.
The question asks: "What number, n, of binary digits is required to represent the same number using 10 trinary digits?"
The solution is:
n . ln(2) = 10 . ln(3)
so
n = 10 . ln(3) / ln(2)
n ~ 15.84962
rounding up to the nearest integer digit gives,
n = 16
hence 16 binary digits are required.
That was very good. Thanks. Need to revise, nay to study my log tables. I never learnt them.
As another commenter pointed out, by knowing log2 and log3 tou can work out alnthe logs to 10, plus some fractions, bar log7.
給我的話我會將3^10變成9^5，然後因爲觀察9^n的性質，最大位的數字是由9->8(9*9=81)->7(9*8=72)->6 ... ->1 然後再重複變成9，每9個一個循環，而在9^5情況下，就有5位（因爲沒有到最大位=1的情況），所以目標就是讓2^n成爲5位數。那因爲衆多周知2^10=1024，而2^3=810，所以2^10*2^4就是最小可能，所以n最小是14，然後因爲16*8>100 16*4
2^n could be 16384, 32768 and 65536 in this case.
I don't know log 2, but I do know my powers of 2, so 16384 through 65536 (14 through 16) have 5 digits. Wonder what the best way to figure out log 3 is if you don't know it.
always useful to have both
¹⁰log 2 and ¹⁰log 3 by heart . they give you directly the log values of
4 and 8, of 6 and 9, of 5 (= 1 - log2)
of 1½ (= log3 - log2) ,
and 2½ (= 1 - 2log2) .
just missing log7 here .
Could you share trick how to find x if ax⁴+bx³+cx²+dx+e=0 ?
Assuming n is an integer, n must be one of {14, 15, 16}.
Explanation for a 5-year old: 3^10 clearly has 5 decimal digits. You can evaluate it by repeated multiplication by 3. Now you only need to try the integral powers of 2 to see which one also has exactly 5 decimal digits.
(Now I could delve into the fact that 2^n is a strictly increasing function of n and prove that there cannot be other values for n as a natural number, but that would probably be a bit too much for a 5y-old.)
Note: One thing I would definitely NOT do is to show my 5y-old how to compute (or rather approximate) the decimal logarithm of 3. Because it is totally irrelevant for answering the question.
I love this stuff!
Where are you?
nice
dang u guys memorize logs
👍👍👍👍