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Step-by-Step Solution
Step 1: Understand the Arithmetic Progression (AP) Condition
We are given three terms in AP:
\log_3(2), \quad \log_3\bigl(2^x - 5\bigr), \quad \log_3\bigl(2^x - \tfrac{7}{2}\bigr).
For three numbers a, b, c to be in an arithmetic progression, the middle term must be the average of the first and the third:
2\,\log_3\bigl(2^x - 5\bigr) \;=\; \log_3(2) \;+\; \log_3\Bigl(2^x - \frac{7}{2}\Bigr).
Step 2: Combine the Logarithms on the Right-hand Side
Using the logarithm rule \log(a) + \log(b) = \log(ab) , we get:
2\,\log_3\bigl(2^x - 5\bigr) \;=\; \log_3\Bigl( 2 \,\bigl(2^x - \tfrac{7}{2}\bigr)\Bigr).
Thus,
2\,\log_3\bigl(2^x - 5\bigr) \;=\; \log_3\Bigl( 2\,\bigl(2^x - \tfrac{7}{2}\bigr)\Bigr).
Step 3: Convert the Left-hand Side into a Single Logarithm
Using the property n \,\log(a) = \log(a^n) , we rewrite the left-hand side:
\log_3\Bigl(\bigl(2^x - 5\bigr)^2\Bigr) \;=\; \log_3\Bigl( 2\,\bigl(2^x - \tfrac{7}{2}\bigr)\Bigr).
This implies:
\bigl(2^x - 5\bigr)^2 \;=\; 2\Bigl(2^x - \tfrac{7}{2}\Bigr).
Step 4: Substitute t = 2^x and Simplify the Equation
Let t = 2^x . Then the equation becomes:
(t - 5)^2 = 2\Bigl(t - \tfrac{7}{2}\Bigr) \;=\; 2t - 7.
Expand and simplify:
(t - 5)^2 = t^2 - 10t + 25.
So
t^2 - 10t + 25 = 2t - 7,
which simplifies to
t^2 - 12t + 32 = 0.
Step 5: Solve the Quadratic Equation
The quadratic factors as:
t^2 - 12t + 32 = (t - 4)(t - 8) = 0.
Hence,
t = 4 \quad \text{or} \quad t = 8.
Recalling that t = 2^x :
If t = 4 , then 2^x = 4 \implies x = 2.
If t = 8 , then 2^x = 8 \implies x = 3.
Step 6: Check for Validity in the Original Logarithms
The expressions inside each logarithm must remain positive:
2^x - 5 > 0 \;\Rightarrow\; 2^x > 5.
2^x - \tfrac{7}{2} > 0 \;\Rightarrow\; 2^x > \tfrac{7}{2}.
For x = 2 :
2^2 = 4, \quad 4 - 5 = -1 < 0,
which is not valid for the logarithm \log_3(2^x - 5).
Therefore, x = 2 is rejected.
Step 7: Conclude the Final Answer
The only valid solution is:
x = 3.
