Given that the positive numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are in Geometric Progression (G.P.). This means that there is a common ratio such that:
[tex]\[ b^2 = a \cdot c \][/tex]
We need to determine the nature of the progression of [tex]\(\log_a 2\)[/tex], [tex]\(\log_b 2\)[/tex], and [tex]\(\log_c 2\)[/tex].
First, let's denote the logs as follows:
[tex]\( x = \log_a 2 \)[/tex], [tex]\( y = \log_b 2 \)[/tex], and [tex]\( z = \log_c 2 \)[/tex].
Using the change of base formula for logarithms, we can express each logarithm in terms of base 2:
[tex]\[ x = \frac{\log 2}{\log a} \][/tex]
[tex]\[ y = \frac{\log 2}{\log b} \][/tex]
[tex]\[ z = \frac{\log 2}{\log c} \][/tex]
Given [tex]\( b^2 = a \cdot c \)[/tex], we can also rewrite this in terms of logarithms. Taking the logarithm (base 2) of both sides:
[tex]\[ \log_2 (b^2) = \log_2 (a \cdot c) \][/tex]
Using the properties of logarithms:
[tex]\[ 2 \log_2 b = \log_2 a + \log_2 c \][/tex]
This can be rewritten as:
[tex]\[ 2 \log b = \log a + \log c \][/tex]
Dividing everything by [tex]\(\log 2\)[/tex]:
[tex]\[ 2 \frac{\log b}{\log 2} = \frac{\log a}{\log 2} + \frac{\log c}{\log 2} \][/tex]
Which simplifies to:
[tex]\[ 2y = x + z \][/tex]
This matches the defining property of an Arithmetic Progression (A.P.), which states that for three numbers [tex]\(x, y,\)[/tex] and [tex]\(z\)[/tex] to be in A.P., the middle term [tex]\(y\)[/tex] must be the average of the first and the last term:
[tex]\[ 2y = x + z \][/tex]
Therefore, [tex]\(\log_a 2\)[/tex], [tex]\(\log_b 2\)[/tex], and [tex]\(\log_c 2\)[/tex] are in Arithmetic Progression (A.P.).
The correct answer is:
(1) A.P.
