An example of a logarithmic equation that cannot be solved is log(10 - x^2) = 5log(10 - x^2) = 5
=> 10 - x^2 = 10^5
=> x^2 = 10 - 10^5
10 - 10^5 is a negative number. For this to happen x has to take on a complex value.
The logarithmic equation log(10 - x^2) = 5 is an example of an equation that does not have a real solution.