If a, b and c are positive real numbers such that a > 10 ≥ b ≥ c
and
log₈(a + b) log₂₇(a – b)
------------ + --------------
log₂ c log₃ c
= 2/3 ,
then the greatest possible integer value of a is <
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Explanatory Answer
log8(a+b)/log2c
= {log(a+b)/log8} / {logc/log2}
= (1/3) logc(a+b)
log27(a−b)/log3c
= {log(a−b)/log27} / {logc/log3}
= (1/3) logc(a−b)
(1/3) logc(a+b) + (1/3) logc(a−b)
= (1/3) logc{(a+b)(a−b)}
= (1/3) logc(a²−b²)
Given: (1/3) logc(a²−b²) = 2/3
⇒ logc(a²−b²) = 2
⇒ a² − b² = c²
So, a² = b² + c²
Since a > 10 ≥ b ≥ c,
a should be an integer.
The greatest possible integer value of a is possible if b = c = 10.
Then:
a² ≤ 100 + 100 = 200
⇒ a ≤ √200 ≈ 14.14
∴ The maximum integer value of a = 14
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