The value of a^ _b c -c^ _b a , where a, b, c>0 but a, b, c 1 , is

Step-by-step solution

Let x = a^ _ b c . Taking the logarithm with base b on both sides, we get: _ b x = _ b (a^ _ b c ) Using the property _ b (m^n) = n _ b m , we have: _ b x = ( _ b c) ( _ b a) Similarly, let y = c^ _ b a . Taking the logarithm with base b on both sides, we get: _ b y = _ b (c^ _ b a ) _ b y = ( _ b a) ( _ b c) Since _ b x = _ b y , it follows that x = y . Therefore, a^ _ b c - c^ _ b a = x - y = 0 . Answer: 0