Martian logs
Source: 2019 AIME II #6
March 22, 2019
AMCAIMElogarithmsalgebrasystem of equations2019 AIME II2019 AMC
Problem Statement
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base , for some fixed . A Martian student writes down
\begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54
\end{align*}
and finds that this system of equations has a single real number solution . Find .