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DIFFICULT logarithms question (non-calculator)

To nine decimal places log2 = 0.3010299996 and log3 = 0.477121255.
(i) calculate log5 and log 6 to three decimal places.(no calculator)
(ii) by taking logs, or otherwise, show that 5* 10^47<3^100<6*10^47
(iii) hence write down the first digit of 3^100
(iv) find the first digit of each of the following numbers:2^1000, 2^10000 and 2^100000
I must stress that all these questions are difficult only because you can't use a calculator
If you can do these questions, I hereby declare that you're very clever
Reply 1
Avatar for Ano123
Ano123
15
Now this is a STEP question for sure. STEP 1 I believe.
ii) log⁡5+47<100log⁡3<log⁡6+47 \log 5 + 47 <100\log 3 < \log 6 + 47 , taking 47 away and using answers in first part of the question it is clear that the given double inequality is true.
(iii) Easy
For the rest it is important to remember that you can find the decimal approximations of the log of the numbers, 2, 3, 4, 5, 6, 8 and 9. So as long as none of them start with the digit 7 (which they won't) you're fine.
Original post by Ano123
Now this is a STEP question for sure. STEP 1 I believe.
ii) log⁡5+47<100log⁡3<log⁡6+47 \log 5 + 47 <100\log 3 < \log 6 + 47 , taking 47 away and using answers in first part of the question it is clear that the given double inequality is true.
(iii) Easy
For the rest it is important to remember that you can find the decimal approximations of the log of the numbers, 2, 3, 4, 5, 6, 8 and 9. So as long as none of them start with the digit 7 (which they won't) you're fine.


A* student!

Not many people can do this question although I had a good time solving it

R u doing step this year or next year?
Reply 3
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Ano123
15
Original post by STRANGER2
A* student!

Not many people can do this question although I had a good time solving it

R u doing step this year or next year?


This year, only STEP I though.
Original post by Ano123
This year, only STEP I though.


Wat year u in?
Reply 5
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Ano123
15
Original post by STRANGER2
Wat year u in?


11
Original post by Ano123
11


when did u do a level maths?
Reply 7
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Ano123
15
Original post by STRANGER2
when did u do a level maths?


Currently doing it.

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