Logarithms Quiz

$10^4 = 10000$, so $\log_{10} 10000 = 4$.

$\log a + \log b = \log(ab)$. Thus $\log(ab) = 3 \Rightarrow ab = 10^3 = 1000$.

$4x = 2^6 \Rightarrow 4x = 64 \Rightarrow x = 16$.

$5^3 = 125$, so $\log_5 125 = 3$.

$\log_{10} 3 \approx 0.4771$. This is a standard value used in competitive exams.

$\log_2 10 = \log_{10} 10 / \log_{10} 2 = 1 / 0.3010 \approx 3.32$.

$\log(x^2 - 1) = 1 \Rightarrow x^2 - 1 = 10 \Rightarrow x^2 = 11 \Rightarrow x = \sqrt{11}$.

$50 \times 0.3010 = 15.05$. Number of digits = $\lfloor 15.05 \rfloor + 1 = 16$.

$\log(1000^{1/2}) = 1/2 \log(1000) = 1/2 \times 3 = 1.5$.

$x = 3^4 = 81$.