Answer:
x = log₂(3) ≈ 1.5849625
Step-by-step explanation:
We want to find the solution to an exponential equation that is in the form of a quadratic.
Setup
Let z = 2^x. Then the equation becomes ...
z^2 -z -6 = 0 . . . . . a quadratic relation in z
Solution
This quadratic can be factored to find the solutions for z:
(z -3)(z +2) = 0
Values of z that make this true are ...
z = 3 and z = -2
We know that 2^x = -2 is impossible for real values of x, so the solution z=-2 is extraneous. The useful solution is ...
z = 3 = 2^x
Taking the logarithm, base 2, we have ...
x = log₂(3) ≈ 1.5849625
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Additional comment
The logarithm gives a complex result for negative arguments. The other solution is ...
x = log₂(-2) ≈ 1 +4.5323601i
The imaginary part is π/ln(2).
The attachment shows a graphing calculator solution. The same calculator can provide an iterated numerical solution without much additional effort. It gives the same answer as above.
