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You can also ask for a numerical apriximation via the funcion/method vumerical_approx (with handy abbreviations n and N) ; in this case, this yelds a complex value with very small imaginary part, which Sage has trouble proving null but probably is:

Let's call Sol your solution. Then :

sage: Sol.rhs().imag_part().log().n()
sage: Sol.rhs().imag_part().log().n(digits=30)
sage: Sol.rhs().imag_part().log().n(digits=60)

Trying to obtain your solution by other means (e. g. as a root of a polynomial in an exact ring) may enable Sage to prove that thos solution is real.