KCC's Quizzes AQQ253 about a funny equation

8^x+2^x=130

(2³)^x+2^x=130

(2^x)³+2^x=130

Let a=2^x which results in the cubic equation:

a³+a-130=0

By inspection, the real root is a=5.  The two other roots are non-real complex conjugate numbers.  See Cardano’s formula in Wikipedia for the calculation of the complex roots.

X=log(a)/log(2)=2.321928095

8^x+2^x=130

(2³)^x+2^x=130

(2^x)³+2^x=130

Let a=2^x which results in the cubic equation:

a³+a-130=0

By inspection, the real root is a=5.  The two other roots are non-real complex conjugate numbers.  See Cardano’s formula in Wikipedia for the calculation of the complex roots.

X=log(a)/log(2)=2.321928095

Correct! That equation accepts 3 solutions: one real and 2 complex. Congratulations

If you calculate x from the complex solutions you express the numbers as |mag| times e^(iTheta).  Then you use the natural log (ln) to solve for x.  This results in an infinite number of answers separated by 2Pi.

Indeed!