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What Is The First Digit of 2^100000 ?

Kolya

17

What is the first digit of $2^{100000}$ given $log_{10} 2 = 0.301029996$ and $log_{10} 3 = 0.47712125$ ?

Reply 1

arran

1 repeating digits 2,4,8,1,3,6,1,2,5,1 every 10th digit is a 1 tho this doesnt answer the q...
EDIT: and is wrong as it all goes to pot relatively quickly... damnabbit

Reply 2

KAISER_MOLE

7

so 100000log2 = 30102.9996 blah

log(8x10³⁰¹⁰²) is slightly less than this

log(9x10³⁰¹⁰²) is slightly less than this also (how you can get this from only log2 value though? )

So I guess the first digit as 9

Reply 3

Kolya

OP

17

Sorry, a clever person already solved it for me :smile:

If you are interested, you also get the value of log 3 (edited original question). I had exactly the same idea arran but it is incorrect :frown:

Reply 4

danniella

4

Write out log x for x from 1 to 10 (luckily you don't need 7 and you can do the rest easily)
log 2^1000 = 1000 log 2 = 301.029996
so 2^1000 = 10^301 x 10^0.29996
and 0<0.029996<0.301
so 10^0 < 10^0.29996 < 10^0.301
so the first digit is a 1.
basically, the decimal bit is the number that needs to be multiplied by the power of 1.

so 2^100000, then...
log 2^100000 = 100,000 log 2
= 30102.9996
= 10^30102 x 10^0.9996

and 0.954 < 0.9996 < 1
so the first digit is a 9

(using logs as "x=log 5, so 10^x = 5")

love danniella.

Reply 5

KAISER_MOLE

7

ah, you also get log3? Magical :p:

just have a splash around with 2log3 then

Reply 6

KAISER_MOLE

7

Nice work danniella - too nice if you ask me, get back to your GCSE revision!!

Reply 7

danniella

4

KAISER_MOLE

Nice work danniella - too nice if you ask me, get back to your GCSE revision!!

*grumbles*
Okay, okay, I'm going... Non-calculator on Monday... wish me luck!
:rolleyes:

love danniella

Edited to add: yes, you were given log 2 and log 3 in the question.

Reply 8

Kolya

OP

17

KAISER_MOLE

Nice work danniella - too nice if you ask me, get back to your GCSE revision!!

Dem young whippersnappers eh? :p:

Reply 9

Popa Dom

2

I think that question was in my Oxford admissions exam this year, but I think it was a smaller power and they also asked for the number of digits. Not that I could do that one, it always makes you feel better about yourself when a GCSE student can do Alevel maths better than you can...

Reply 10

yusufu

16

Popa Dom

I think that question was in my Oxford admissions exam this year, but I think it was a smaller power and they also asked for the number of digits. Not that I could do that one, it always makes you feel better about yourself when a GCSE student can do Alevel maths better than you can...

I've seen this question in the STEP booklet as well.

Reply 11

meef cheese

10

Yeah, I remember that question from the test. Funny thing is I got it right then but couldn't remember how to do it. I really do think I'm getting worse at Maths...

Reply 12

Kolya

OP

17

For the record it is Q1 on STEP I from 2000 (not in the booklet though?). The handwritten solutions that are floating around on the web do not show how to answer the question.

Interesting to hear a similar one was on the Oxford test...

Reply 13

insparato

15

You guys ever heard of log books? Isnt this kinda question based on what log books were about. My dad had a 5 figure log book :P.

Edit: Danniella maybe you should be doing a maths degree rather than medicine :p: