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

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

### 2 Answers by Expert Tutors

Roman C. | Masters of Education Graduate with Mathematics ExpertiseMasters of Education Graduate with Mathe...
4.9 4.9 (332 lesson ratings) (332)
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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.

Anthony P. | Experienced tutor in earth sciences and basic math to trigonometry Experienced tutor in earth sciences and ...
5.0 5.0 (67 lesson ratings) (67)
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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.