Given $10^3=1000,10^4=10000,2^{10}=1024,2^{11}=2048,2^{12}=4096,2^{13}=8192$,what are the largest $a$and smallest $b$ such that $a < \log_{10} 2 < b$

Question

If one uses only the information $10^3=1000,10^4=10000,2^{10}=1024,2^{11}=2048,2^{12}=4096,2^{13}=8192$,what are the largest $a$ and smallest $b$ such that one can prove $a < \log_{10} 2 < b$

I am kind of clueless about this one. It's not clear to me if I have to find $a,b$ in terms of logarithms and what is their relation with the given data.

By checking up with a calculator I see that $\log 2 \approx 0,3 $ but I don't see how I can now get a bound for this value given the information above.

$2^{13}<10^4\iff 13\log_{10}2<4\iff \log_{10}2<\frac{4}{13}$, because $\log_{10}x$ is a strictly increasing function. — user236182, Dec 31 '15 at 18:46

score 3 · Accepted Answer · answered Dec 31 '15 at 18:41

3

The logarithm function is monotonic increasing so from $10^3\lt 2^{10}$ you can take logarithms to base $10$ and obtain $3\lt \log_{10} 2^{10}=10\log_{10}2$ which gives you a lower limit of $0.3$. Now do your best with the upper limit.

answered Dec 31 '15 at 18:41

Mark Bennet

100,194

Got it.Thanks for the help. – Mr. Y Dec 31 '15 at 18:49

Given $10^3=1000,10^4=10000,2^{10}=1024,2^{11}=2048,2^{12}=4096,2^{13}=8192$,what are the largest $a$and smallest $b$ such that $a < \log_{10} 2 < b$

1 Answers1