# How to solve this system with logarithms?

I mean a symbolic solution of $$\frac{\log \left(x^2+10 x+25\right)}{\log (y-10)}+\frac{\log \left(y^2-20 y+100\right)}{\log (x+5)}=4\land \frac{\log (2 x+9)}{\log (y-10)}+\frac{\log (23-2 y)}{\log (x+5)}=2$$ over the reals. The Reduce command fails with it:

Reduce[Log[x + 5, y^2 - 20 y + 100] + Log[y - 10, x^2 + 10 x + 25] == 4 &&
Log[x + 5, -2 y + 23] + Log[y - 10, 2 x + 9] == 2, {x,y}, Reals]


Reduce::nsmet: This system cannot be solved with the methods available to Reduce.

The ContourPlot command performs only one curve.

ContourPlot[{Log[x + 5, y^2 - 20 y + 100] +
Log[y - 10, x^2 + 10 x + 25] == 4, Log[x + 5, -2 y + 23] + Log[y - 10, 2 x + 9] == 2},
{x, -4.5, 10}, {y, 10, 11.5}, WorkingPrecision -> 15, PlotPoints -> 50]


Numerical commands fail too.

NSolve[Log[x + 5, y^2 - 20 y + 100] + Log[y - 10, x^2 + 10 x + 25] ==  4 &&
Log[x + 5, -2 y + 23] + Log[y - 10, 2 x + 9] == 2, {x, y}]


NSolve::nsmet: This system cannot be solved with the methods available to NSolve.

FindRoot[Log[x + 5, y^2 - 20 y + 100] +
Log[y - 10, x^2 + 10 x + 25] == 4 &&
Log[x + 5, -2 y + 23] + Log[y - 10, 2 x + 9] == 2, {{x, 5}, {y,10.5}}, AccuracyGoal -> 3]


FindRoot::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations. {x -> -3.83392*10^101 - 1.89741*10^97 I, y -> 2.84399*10^101 - 1.92551*10^97 I}

• FindRoot[{one, two}, {{x, -3.9}, {y, 11.2}}]delivers a result (one and two are your equations) – mgamer Oct 22 '18 at 9:53
• @mgamer: Thank you. It is useful. – user64494 Oct 22 '18 at 10:05
• An exact solution is $\left\{ x=-{\frac{22}{5}},y={\frac{53}{5}} \right\}$. – user64494 Oct 22 '18 at 15:50