find x in the equation x=2^x+6

Question

solve for the value of x in the equation x= 2^x+6

 

Answer 1

If there was a solution, it would have to be positive, since RHS is always positive.

The average slope of RHS from x = -1 to 0 is 32 and the slope strictly increases as x increases. Thus the slope of RHS is greater than 32 at all x > 0 so the RHS increases faster than x and RHS > x at x=0. Thus there can't be any intersection.

If you are taking calculus, then there is another way:

Let y = 2^x+6 - x. Then dy/dx = 2^x+6*ln 2 - 1. Setting dy/dx = 0 gives.

2^x+6*ln 2 = 1

2^x+6 = 1/ln 2

x+6 = -log₂ ln 2

x = -6 - log₂ ln 2

At this point, y = (1/ln 2) + 6 + log₂ ln 2 > 0.

d²y/dx² = 2^x+6 (ln 2)² > 0 for all x so y = (1/ln 2) + 6 + log₂ ln 2 is a global minimum and so 2^x+6 > x for all real x. Thus there is no solution.

 

Answer 2

There is no solution to this equation. Substituting any possible solution for X results in a false statement. For example, suppose X = 1, that would give 1 = 2⁶, which is clearly not true.