Given: 2^a = 3^b = 12^c



To find: ab - c(a + 2b)



Solution:


Let's start by finding out the values of a, b, and c using the given equation.



We know that 2^a = 3^b, which means that a log 2 = b log 3 (using logarithmic property). Taking c as the common power, we can say that:


2^a = 3^b = (2^2 * 3)^c = 2^(2c) * 3^c


Now we can equate the two equations to get:


2^a = 2^(2c) * 3^c


Taking log on both sides, we get:


a = 2c log 2 + c log 3



Similarly, we can equate 3^b = 2^(2c) * 3^c to get:


b = c log 2 + 2c log 3



Now we can substitute the values of a and b in the expression ab - c(a + 2b) to get:


ab - c(a + 2b) = (2c log 2 + c log 3) * (c log 2 + 2c log 3) - c(2c log 2 + 4c log 3)


Simplifying this expression, we get:


ab - c(a + 2b) = c^2(log 2)^2 - c^2(log 3)^2



Therefore, the value of ab - c(a + 2b) is c^2(log 2)^2 - c^2(log 3)^2.