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 $\dfrac{40}{132}$

Answer: C

Concepts:logarithm inequality bounding to limit cases

Difficulty rating: 1710

Small Hint:

Compare $10^3$ with $2^{10}$ for a lower bound

Big Hint:

Compare $2^{13}$ with $10^4$ for an upper bound

Solution:

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.

1967 AMC 12 Problem 26

Answer: C

Difficulty rating: 1710

Small Hint:

Big Hint:

Solution:

Problem 26 in Other Years