# Solve equation using Mathematica

I want to solve this equation using Mathematica:

$$4^x -18^x-8=0$$

I tried this code and it didn't work:

Solve[4^x-18^x-8==0,x]


• Have you read the error message? "This system cannot be solved with the methods available to Solve." Jul 31, 2019 at 8:21
• There are no real solutions as 4^x - 18^x < 1 for real x. Do you want a complex answer? Jul 31, 2019 at 8:32
• Yes this was the error message Jul 31, 2019 at 8:32
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• @Sadisharandinu, Try to see and find the roots (whether Real or Complex) using Contour plot: mathematica.stackexchange.com/a/194654/44142 Aug 1, 2019 at 5:06

You just need to add a domain restriction. For example:

Solve[4^x - 18^x - 8 == 0 && Abs[x] < 10, x] //N


{{x -> 0.655338 - 9.68327 I}, {x -> 0.655338 + 9.68327 I}, {x -> 0.682744 - 5.32247 I}, {x -> 0.682744 + 5.32247 I}, {x -> 0.712648 - 0.969253 I}, {x -> 0.712648 + 0.969253 I}, {x -> 0.742883 - 3.37599 I}, {x -> 0.742883 + 3.37599 I}, {x -> 0.771278 + 7.71356 I}, {x -> 0.771278 - 7.71356 I}}

(I used N to convert the exact values (which are Root objects) into complex number). You can also use Re and Im:

Solve[4^x - 18^x - 8 == 0 && 0<Re[x]<10 && 10<Im[x]<20, x] //N


{{x -> 0.615189 + 18.4231 I}, {x -> 0.632332 + 14.0506 I}, {x -> 0.795925 + 12.0444 I}, {x -> 0.815342 + 16.3697 I}}

If you can live with numerical complex-valued solutions,

FindRoot[4^x - 18^x - 8 == 0, {x, I}]
(*    {x -> 0.712648 + 0.969253 I}    *)

FindRoot[4^x - 18^x - 8 == 0, {x, -I}]
(*    {x -> 0.712648 - 0.969253 I}    *)


There are infinitely many more solutions. To look for them, you could use starting points gathered from a DensityPlot or similar.