For what value(s) of x will....

Question

For what value(s) of x will 4 x 2^x = 2^6x+3?

Answer 1

4 * 2^x = 2^(6x) + 3

4 * 2^x = (2^x)^6 + 3

0 = (2^x)^6 - 4 * (2^x) + 3

0 = Z^6 - 4*Z + 3 where Z = (2^x)


Z=1 is a solution as 1^6 - 4(1) + 3 = 1 - 4 + 3 = -3 + 3 = 0

Then Z-1 is a factor.
By Synthetic division:
(Z-1)(Z^5 + Z^4 + Z^3 + Z^2 + Z - 3) = 0


The 5th degree polynomial does not have any rational roots.
The only possible rational roots are { +- 1, +- 3}
and none of them work.

So it must be solved numerically.
The root is approximately Z=0.8343

factoring it out using numeric synthetic division

(Z - 0.8343)( Z^4 + 1.8343 Z^3 + 2.53035649Z^2 + 3.111076419607Z + 3.5955710568781201)

The root of the 4th degree polynomial is

Z=.6316

factoring it out using numerical synthethic division:

(Z-0.8343)(Z - 0.6316) (Z^3 + 2.4659Z^2 + 4.08781893Z + 5.692942436188)

The root of the cubic polynomial is Z= 0.8329


(Z-0.8343)(Z - 0.6316)(Z-0.8329) (Z^2 + 3.2988X + 6.83538945)

(Z-0.8343)(Z - 0.6316)(Z-0.8329) (Z^2 + 3.2988Z + 6.83538945)

The roots of the quadratic are Z=1.442 and Z=-4.741

In summary the six roots are Z={1, 0.8343, 0.6313, 0.8329, 1.442, -4.741 }

Solving Z = 2^x for X ---> logZ = x , where the log is base 2

So the five roots are:

log 1, log 0.8343, log 0.6313, log 0.8329, log 1.442, log -4.741

which are:
0
-0.261361849
-0.663602344
-0.263784802
0.528071165

There are five roots, not six, because logs of negatives are undefined.
So one root is lost.