The solution is x=0.464343 approx. using the Intermediate Value Theorem.
METHOD
The quantity on the right can only be positive when 0<x<1. So the solution must be between these limits.
The domain can be split into 0.1 intervals, and by substituting x=0.1 to 0.9 in the expression log(6x+10)-log(x)/log(½) where log is assumed to be base 10, it can be seen that the expression changes sign between 0.4 and 0.5.
This interval can be split into hundredths: 0.40, 0.41, ..., 0.49, 0.50 and re-evaluated. By continuing this process an approximate value of x can be found.
If the logs are natural logs x=0.188211 approx.
Note that 1/log₁₀(½)=log[base ½](10), so the question can be expressed:
log(6x+10)=log(x)log[base ½](10) and log[base ½](x)=-log₂(x).