# Numerical solution to given equation involving one variable

Any neat way to solve this equation?

NSolve[(8^x - 2^x)/(6^x - 3^x) == 2, x]

NSolve[(8^x - 2^x)/(6^x - 3^x) == 2, x,
Method -> {"UseSlicingHyperplanes" -> False}]


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

• FindInstance[(8^x - 2^x)/(6^x - 3^x) == 2 && x > 0, x, Reals, 1]
– I.M.
Dec 1, 2022 at 5:24
• NSolve[(8^x - 2^x)/(6^x - 3^x) == 2, Reals] Dec 1, 2022 at 8:16

 Plot[{(8^x - 2^x)/(6^x - 3^x), 2}, {x, -3, 3}]


So you can try to help NSolve a little

 NSolve[(8^x - 2^x)/(6^x - 3^x) == 2 && 0 <= x , x]


NSolve[(8^x - 2^x)/(6^x - 3^x) == 2 && x > -10 && x <= 10, x, Reals]


{{x -> 1.}}

FindInstance[(8^x-2^x)/(6^x-3^x)==2&&x>-10&&x<=10,x,Reals]


{{x -> 1}}

Unfortunately,

FindInstance[(8^x - 2^x)/(6^x - 3^x) == 2 && x > -10 &&   x <= 10, x, Reals, 2]


{{x -> 0}, {x -> 1}}

which is not correct.

• Since Limit[(8^x - 2^x)/(6^x - 3^x), x -> 0]==Log[4]/Log[2], one can redefine (8^x - 2^x)/(6^x - 3^x) at x==0 to make the result of FindInstance correct. – Dec 1, 2022 at 5:56