Basic Mathematics & Logarithm
Logarithmic Inequalities and Bounds
Grade Class 11
Question:
<p>Let \(\log_5 N = I_1 + f_1\) and \(\log_3 N = I_2 + f_2\), where \(I_1, I_2\) are integers and \(f_1, f_2 \in [0,1)\). If \(I_1 = 2\) and \(I_2 = 3\), then the maximum integral value of \(N\) is greater than or equal to</p>
\(79\)
\(80\)
\(81\)
\(82\)
Step-by-Step Solution
Key Concept: Translate the integer parts of the logarithms into interval bounds for $N$. Then intersect the intervals.
Notice that $I_1=2$ means $2 \le \log_5 N < 3$, so $25 \le N < 125$. A clever move here is to do the same with the second condition: $I_2=3$ gives $3 \le \log_3 N < 4$, hence $27 \le N < 81$. Intersecting the two intervals, we get $27 \le N < 81$. So the maximum integral value is $80$. That means it is certainly greater than or equal to $79$ and $80$, but not $81$ or $82$.
Correct Answer: A, B