System of equations with logarithm

Algebra Level 2

If $x=\alpha$ and $y=\beta$ are solutions to the system of equations $\begin{aligned} 2^x-2\cdot 4^{-y} &=7 \\ \log_2 (x-2)-\log_2 y &=1, \end{aligned}$ what is the value of $10\alpha\beta?$

The answer is 15.

1 solution

Arjun Bharat
May 5, 2014

In this sum, after solving the second equation we get x = 2y + 2. Then, after substituting this value in the first equation, we get: - 2^(2y+2) + 2^(1-2y )= 7 2^2y (2^2+ 2^(1-4y) )= 7 Note that y cannot have a negative value, because that would cause the equation to have a fractional value and not a whole number value. Even at y = -1, the equation is not true. So let us assume that y > 0. Note that 7 can also be written as 2^3 - 1. Therefore: 2^(2y+2) + 2^(1-2y ) = 2^3 - 1 Equating 2^(2y + 2) and 2^3, we get y = 0.5. Therefore, x = 3. Therefore, the value of 10αβ = 10 × 1.5 = 15.

System of equations with logarithm

The answer is 15.

1 solution

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