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Algebra Level 2

The value of $\log_{20} 3$ lies between the interval $\left ( \frac 1 n , \frac 1 {n-1} \right )$ for some positve integer $n$ . What is the value of $n$ ?

Details and Assumptions :

This problem can be solved without the use of tables or calculators.

5

4

3

2

1 solution

Caleb Townsend
Apr 4, 2015

$\log_{27} 3 < \log_{20} 3 < \log_{9} 3 \\ \frac{1}{3} < \log_{20} 3 < \frac{1}{2}$ So the logarithm is between $1/3$ and $1/2.$ Comparing $3$ and $2$ to $n$ and $n-1,$ $n=3.$

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