log[4](a)=(1/2)log[2](a) and log[8](b+1)=1/3log[2](b+1); so (1/2)log[2](a)=(1/3)log[2](b+1) (also 3log(a)=2log(b+1); log(a^3)=log((b+1)^2) with log to any base as long as it's the same on each side; so a^3=(b+1)^2).
Now we can easily raise both sides as exponents of 2: sqrt(a)=cuberoot(b+1); a^(1/2)=(b+1)^(1/3).
Raise both sides to the power of 6: a^3=(b+1)^2.