IMO Practice Test — Logarithms
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
hard
If \(\log_2 3 = a\) and \(\log_2 5 = b\), find \(\log_2 15\)
\(a+b\)
\(ab\)
\(a-b\)
\(a/b\)
Explanation: 15=3×5, so log15=log3+log5=a+b
Question 2 of 6
hard
Solve for x: \(\log_5 x = 3\)
15
75
125
625
Explanation: 5³=125
Question 3 of 6
hard
Simplify: \(\log 4 + \log 5 + \log 5\)
\(\log 20\)
\(\log 100\)
\(\log 14\)
\(\log 50\)
Explanation: =log(4×5×5)=log100
Question 4 of 6
hard
If \(\log_{10} 2 = 0.3010\), find \(\log_{10} \sqrt[3]{2}\)
0.1003
0.2007
0.3010
0.9030
Explanation: √[3]{2}=2^{1/3}, log=1/3×0.3010≈0.1003
Question 5 of 6
hard
If \(\log x + \log y = 3\) and \(\log x - \log y = 1\), find \(x\)
10
100
1000
10000
Explanation: Adding: 2log x=4 → log x=2 → x=100
Question 6 of 6
hard
The value of \(\log_3 81 + \log_4 64\) is:
4
5
6
7
Explanation: log₃81=4 (3⁴=81), log₄64=3 (4³=64), sum=7