Question

Logarithm Question

Find the number of zeroes between decimal point and first significant digit of \(\left(\cfrac{1}{3}\right)^{500}\) , if \(\log_{10} 3 = 0.47712\)

Answer 1

This time, I promise, I don't have *any* idea of how to solve it.

Answer 2

@Kainui @dan815 @ganeshie8 @BSwan

Answer 3

turn it into base 10 so u can see how many decimals its moving over

Answer 4

hint :-
\(log_b x= y\)

\(b^y=x\)

Answer 5

\[\frac{1}{3}=3^{-1}\]\[3^{-500}\]\[\log_{10}3=.47712\]\[-500*\log_{10}3=-500*.47712\]\[3^{-500}=10^{-500*.47712}\]

Answer 6

Okay, well, I almost got what @Kainui said. But, I'm really confused that how to get to the answer?

Answer 7

you got it alrready

Answer 8

oh kainui did

Answer 9

Well 10^(-1) means .1 and 10^(-2) means .01, so maybe that helps lead you into the right direction, since powers of 10 are literally decimal places now.

Answer 10

-500*0.477 = -238.5 so 238 decimals over s0 237 zeros

Answer 11

10^{-238.56}

yeah... got it....

Answer 12

Thanks everyone! Specially, @Kainui , @BSwan and @dan815 .

Answer 13

haha i did nothing :P

Answer 14

Hmm, wouldn't it be 239 and 238 zeroes?

Answer 15

238.56 \(\approx\) 239

\(10^{-239} \) ---> gives 238 zeroes between the decimal point and the first significant digit... right?

Answer 16

no

Answer 17

10^{-1} has 0 zeroes between decimal point and first sig fig
10^{-2} has 1 zero between decimal point and first sig fig

238-1 = 237

Answer 18

ok yes

Answer 19

dnt use the sproimation
10^238.56= 10^238 . 10^ 0.56

Answer 20

aproximation*

Answer 21

@ganeshie8 - Yeah, the but what I think is :

\(\boxed{\bf{500 \times 0.47712 = 238.56 \color{blue}{\approx 239}}}\)

So, \(10^{-239} \) has 238 zeroes...

That's what I think.

Answer 22

ya 239 decimal places and 239 zeros

Answer 23

238*

Answer 24

m feeling more dyslexic LOL yes 238 zeroes :)

Answer 25

http://www.wolframalpha.com/input/?i=10%5E-3.6

\[10^{-3.6} \approx .000251\]

So instead of worrying about what the rule is or should possibly be at that really high number, let's just figure out the rule here.

The rule is for our purposes, we round down from a decimal place apparently, that's the number of zeroes.

He got the answer right a while ago, definitely 238.

Answer 26

go with what bswan said, because even if you approximated down it could still have 238 zeros

Answer 27

Okay, yes, the book also says 238. Right!

Answer 28

like for example 10^-238.3 is still 238 zeros

Answer 29

Oh, how ?

\(10^{-238} \times 10^{-0.3} \)

How will it have 238 zeroes?

Answer 30

yep because 10^ -0.3 is still 0.something right

Answer 31

as long as its anything more than 238

Answer 32

even 238.00000000000001

Answer 33

Yeah, but, we needed to find the zeroes between the decimal point and the first significant digit.

Answer 34

My calculator says : \(10^{-0.3} = 0.5011 ... \)

Answer 35

So, if we consider the amount of zeroes : (betn the decimal point and first significant digit)

we will come up with 237 , am I wrong with my concept here?

Answer 36

0.5*0.1=0.05

Answer 37

as long as number < 1 you will have 1 extra 0

Answer 38

Oh! Got it... Thanks Dan.

Answer 39

ok

Answer 40

good :P