Math Problem? I like math but I am totally stumped on this.?

Question

Hi, I'm 12(7th grade), and I need help(or perhaps to answer to) with the following problem:
"Start with the number 1-double it, you get 2-double it, you get 4-double it, you get 8. Starting with the number 1, how many times will you have to double, so that the doubled number is over ten million?"
I'd really appreciate thorough answers and don't answer with useless things like "do it yourself", because I have tried that, and asked my parents. Thanks!

btw, i feel like its one of those questions where the answer is really simple, but i really have no clue. its hard for the 12 year old mind to process this kind of stuff :P

You guys are all right, but theres a simpler method for this.
My 9th grade brother solved this easily.
You find the square root of 10,000,000. Obviously a calculator is needed, but I found the strategy, which will probably get me more credit then getting the answer. Thanks anway everyone =)

Answer 1

This certainly isn't a problem to which I think you "should" be able to answer, as it involves the use of a mathematical technique known as logarithms, which you almost certainly haven't been taught yet.

If you think of multiplication as "adding things lots of times", there's a mathematical technique called "raise to the power of" which is the same as multiplying things lots of times. Now... in the same way that division is like the opposite of multiplication (formally called the inverse), logarithms are the opposite of raising to the power.

To actually calculate the answer to your problem is then just log2(10 000 000), which is just over 23 and a quarter. (log2 represents the logarithm function.) This means you need to double 24 times to get a number *bigger* than ten million.

Logarithms are a concept which aren't taught to a lot of 16 year olds (at least in the UK), so don't worry if this doesn't make a lot of sense at the moment!

Answer 2

Ok so if you think about it, the equation looks like this:

1*(2^n)

n is increasing integers starting from 0.

since each number is doubled everytime, adding another power of 2 is the same as doubling it. (multiplying by 2 doubles something)

so now we need to solve this:

2^n = 10,000,000

n= ln(10,000,000)/ln(2)

n = 23.25

that means that if you double it 23.25, you will reach 10 million. since we want it to be over 10 million, and since n must be an integer, we should round it up to 24.

you must double it 24 times to reach over 10 million.

Answer 3

Without using mathematics which you have not yet studied, the only way to do it is the long way round.

The answer is 24 times, since

2^24 = 16777216, and

2^23 = 8388608.

Hope this helps, Twiggy.

Answer 4

you have to find a number N so 2^N greater than 10.000.000.
I don´t think this is a problem for a 12 years old

As for your parents,if they remember logarithm
N*log 2 = log 10.000.000 = 7
so N= 7/0.30103 and the next integer is 24

Answer 5

I counted 24 times.

Answer 6

2^n > 10^7

then , nlog2 > 7
then n > 7/log2

Answer 7

= 23.25