The number of roots of the equation 2x + 2x-1 + 2x-2 = 7x +7x-1 +7x-2 is
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0
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1
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2
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3
The equation is 2^x(1 + 1/2 + 1/4) = 7^x(1 + 1/7 + 1/49). This simplifies to 2^x(7/4) = 7^x(57/49). Dividing gives (7/2)^x = (7/4) * (49/57) = 343/228. Since (7/2)^x is a strictly increasing function, there is exactly one real solution for x.
Factor out the lowest power of 2 on the left side to get 2^(x-2)(2^2 + 2^1 + 1) = 2^(x-2)(7). Similarly, factor the right side to get 7^(x-2)(7^2 + 7^1 + 1) = 7^(x-2)(57). Dividing both sides by 7 leaves 2^(x-2) = (57/7)^(x-2), which is written as (2/8.142... )^(x-2) = 1. Since the bases are different and positive, this equality holds only when the exponent is zero, so x - 2 = 0. Thus, there is exactly 1 real root for this equation.