If `((log)_a N)/((log)_c N)=((log)_a N-(log)_b N)/((log)_b N-(log)_c N),w h e r eN >0a n dN!=1, a , b , c >0` and not equal to 1, then prove that `b^2=a c`

Explanation

To prove that \( b^2 = ac \) given the equation

\[ \frac{\log_a N}{\log_c N} = \frac{\log_a N - \log_b N}{\log_b N - \log_c N} \]

where \( N > 0 \), \( N \neq 1 \), and \( a, b, c > 0 \) and not equal to 1, we can follow these steps:

...