csat-suneung 2014 Q20

csat-suneung · South-Korea · csat__math-A 4 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms

For a positive real number $x$, let $f ( x )$ and $g ( x )$ be the characteristic and mantissa of $\log x$, respectively. For a natural number $n$, let $a _ { n }$ be the product of all values of $x$ satisfying $f ( x ) - ( n + 1 ) g ( x ) = n$. What is the value of $\lim _ { n \rightarrow \infty } \frac { \log a _ { n } } { n ^ { 2 } }$? [4 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

For a positive real number $x$, let $f ( x )$ and $g ( x )$ be the characteristic and mantissa of $\log x$, respectively. For a natural number $n$, let $a _ { n }$ be the product of all values of $x$ satisfying $f ( x ) - ( n + 1 ) g ( x ) = n$. What is the value of $\lim _ { n \rightarrow \infty } \frac { \log a _ { n } } { n ^ { 2 } }$? [4 points]\\
(1) 1\\
(2) $\frac { 3 } { 2 }$\\
(3) 2\\
(4) $\frac { 5 } { 2 }$\\
(5) 3

Paper Questions

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