Powers of $2$ link to $1$ ?

Algebra Level 4

When writing out some powers of 2, we begin to notice something. $\begin{array}{c} \color{#333333}{2^0 = } \color{#D61F06}{1} \\ \color{#333333}{2^{10} = } \color{#D61F06}{1} \color{#333333}{024} \\ \color{#333333}{2^{20} = } \color{#D61F06}{1} \color{#333333}{048576} \\ \color{#333333}{2^{30} = } \color{#D61F06}{1} \color{#333333}{073741824} \end{array}$ It seems that the first digit of the answer when raising 2 to a multiple of 10 is always a 1.

What is the smallest multiple of 10 which can be the exponent of 2 such that the leading digit of the result is not a 1? If you think that there is no such number, enter -1 as your answer.

The answer is 300.

1 solution

Michael Mendrin
Jul 14, 2018

$2^{10n}=(2^{10})^n=1024^n=(1.024)^n \cdot 10^{3n}$

Since $10^{3n}$ does not affect the 1st digit, we look for what value of $n$ such that $(1.024)^n > 2$

$nLog(1.024)>Log(2)$ , so we first solve

$n=\dfrac{Log(2)}{Log(1.024)} \approx 29.2263..$ , so the answer is $10n=300$ , that is, $2^{300}$ starts with $2$

Yes, making it 1.024 is the key to solving it

Stephen Mellor - 2 years, 11 months ago

As I understand it, these types of problems only became "exotic" as students started using calculators instead of slide rules to write numbers in scientific notation.

Either way, nice solution.

Brian Moehring - 2 years, 11 months ago

One of my worries about creating the problem was that people would just try it on a calculator until they found 290 and 300 rather than using the log method. But then again, only 9 out of 18 attempts have been right at the time of writing

Stephen Mellor - 2 years, 11 months ago

Excellent solution

Ram Mohith - 2 years, 11 months ago

@Michael Mendrin Sir, same way!!!! Just a minor typo in the end........the value of n should be the reciprocal of what you have written........!!!

Aaghaz Mahajan - 2 years, 7 months ago

Thanks for the catch! Fixed.

Michael Mendrin - 2 years, 7 months ago

Powers of 2 2 2 link to 1 1 1 ?

The answer is 300.

1 solution

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Powers of $2$ link to $1$ ?