How Do You Solve Equations with Floor Functions and Logarithms?

[inqusoc]

hi I'm new here. For a couple of days i am struggling to methodically solve an equation involving the integer part of a fraction but nothing yet. The equation is;

I found that the apparent solution is 53 as x HAS to be integer. Yet, I cannot mathematically prove how to compute x to be equal to 53. Any opinion is welcome, thanks!

 

[mfb]

There is no way to solve the equation analytically with x and ln(x) like that. Numerical approaches are the best option, for small integer x this just means trial and error.

52 to 56 all work for x.