Question

Find a rational number which is $50$ times its own logarithm to the base $10$ is

Answer 1

Given that, $a=50{\log }_{10}a$
Since, $a$ is a rational number.
Then put $a={10}^k$, where $k$ is a rational number.
${10}^k=50{\log }_{10}{10}^k$
$\Rightarrow {10}^k=50k$
Only possible value of $k$ is $2$
Therefore, $a={10}^2=100$
Ans: 100