1967 AMC 12 Problem 26

Question

If one uses only the tabular information

10^3=1000,

10^4=10{,}000,

2^{10}=1024,

2^{11}=2048,

2^{12}=4096,

2^{13}=8192,

then the strongest statement one can make for

\log_{10}2

is that it lies between:

\dfrac3{10}

and

\dfrac4{11}

\dfrac3{10}

and

\dfrac4{12}

\dfrac3{10}

and

\dfrac4{13}

\dfrac3{10}

and

\dfrac{40}{132}

\dfrac3{11}

and

\dfrac{40}{132}

Compare

10^3

with

2^{10}

for a lower bound

Compare

2^{13}

with

10^4

for an upper bound

From

10^3\lt2^{10}

we get

3\lt10\log_{10}2,

so

\log_{10}2\gt\frac{3}{10}.

From

2^{13}\lt10^4

we get

13\log_{10}2\lt4,

so

\log_{10}2\lt\frac{4}{13}.

This is the narrowest listed interval justified by the table.

Therefore, the correct answer is C.

Problem 26 in other years