Solving non-linear equations of four variables.

Question

I am not sure if I can find explicit expressions for a, b, c and d (all of these positive)from the following system of equations:

$ a \log(2b) -\frac{a}{2} \log((1 +2b)^2+4)- (2c)^d-\log(5) =0$

$ a \log(2b) -\frac{a}{2} \log((4 +2b)^2+9)- (3c)^d-\log(6)=0$

$ a \log(2b) -\frac{a}{2} \log((9 +2b)^2+16)- (4c)^d-\log(7) =0$

$ a \log(2b) -\frac{a}{2} \log((16 +2b)^2+25)- (5c)^d-\log(8)=0$

Answer 1

There is a solution somewhat near to $$ (a = -0.2323, b = +0.0011, c = 6.6481, d = - 7.3427) $$ but that is just a numerical approximation obtained by minimizing the squared error terms on the four equations.

For this, I assumed all the variables were real, and $b$ and $c$ positive.