Search 75,788 tutors
FIND TUTORS
Ask a question
0 0

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

Tutors, please sign in to answer this question.

2 Answers

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

2x+6*ln 2 = 1

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

 

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.