If 1, log₁₀(4ˣ - 2) and log₁₀(4ˣ + 18/5) are in arithmetic progression for a real number x, then the value of the determinant |[2x, x-1, x²], [1, 0, x], [x, 1, 0]| is equal to: ______.
If 1, log₁₀(4ˣ - 2) and log₁₀(4ˣ + 18/5) are in arithmetic progression for a real number x, then the value of the determinant |[2x, x-1, x²], [1, 0, x], [x, 1, 0]| is equal to: ______.
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1 Answer
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The numbers 1, log10(4^x - 2), and log10(4^x + 18/5) are in an Arithmetic Progression (A.P.).
This means that the corresponding numbers 10^1, 10^(log10(4^x - 2)), and 10^(log10(4^x + 18/5)) are in a Geometric Progression (G.P.).
So, 10, 4^x - 2, and 4^x + 18/5 are in G.P.For a G.P., the square of the middle term is equal to the product of the other two terms:
(4^x - 2)^2 = 10 * (4^x + 18/5)
Let y = 4^x.
(y - 2)^2 = 10y + 36
y^2 - 4y + 4 = 10y + 36
y^2 - 14y - 32 = 0
(y - 16)(y + 2) = 0
So, y = 16 or y = -2.Since y = 4^x, y must be positive. Thus, 4^x = 16, which gives x = 2.
The determinant calculation that follows appears to be unrelated to the
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