CAT 2023 Slot 3 QA Question 3: For a real number x, if $\dfrac{1}{2}$, $\dfrac{\log_3(2^x - 9)}{\log_3 4}$, and $\dfrac{\log_5\left(2^x + \dfrac{17}{2}\right)}{\log_5 4}$ are in an arithmetic progression, then the common difference

Question

For a real number x, if $\dfrac{1}{2}$, $\dfrac{\log_3(2^x - 9)}{\log_3 4}$, and $\dfrac{\log_5\left(2^x + \dfrac{17}{2}\right)}{\log_5 4}$ are in an arithmetic progression, then the common difference

Accepted Answer

Using change of base, the second and third terms can be written with base 4.

$\dfrac{\log_3(2^x - 9)}{\log_3 4} = \log_4(2^x - 9)$
$\dfrac{\log_5\left(2^x + \dfrac{17}{2}\right)}{\log_5 4} = \log_4\left(2^x + \dfrac{17}{2}\right)$

For three terms in AP, twice the middle term equals the sum of the