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What is solution in real number of equation (1/2)^x=2^(x-2)?

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yapayapa | (Level 2) Honors

Posted July 24, 2013 at 1:41 PM via web

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What is solution in real number of equation (1/2)^x=2^(x-2)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 24, 2013 at 2:33 PM (Answer #1)

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You should use negative power property, such that:

`(1/2)^x = 2^(-x)`

Replacing `2^(-x)` for `(1/2)^x` yields:

`2^(-x) = 2^(x-2)`

Equating the corresponding exponents yields:

`-x = x - 2 => -x - x = -2 => -2x = -2 => x = 1`

Sketching the graphs of the functions `y = 2^(-x)` and `y = 2^(x-2)` , you may notice that they intersect at `x = 1` and `y = 1/2.`

Hence, evaluating the solution to the given exponential equation, using the properties of exponents, yields `x = 1.`

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