To solve the equation (a / b)x - 1 = (b / a)x - 3, we can start by simplifying both sides of the equation.

Step 1: Simplify the left side of the equation
(a / b)x - 1 = (b / a)x - 3
Multiply both sides of the equation by b to eliminate the fraction:
a^x - b = (b^2 / a)x - 3b

Step 2: Simplify the right side of the equation
(b / a)x - 3 = (b^2 / a)x - 3
Multiply both sides of the equation by a to eliminate the fraction:
ab^x - 3a = b^2x - 3a

Step 3: Combine like terms
a^x - b = ab^x - 3a
Rearrange the terms:
a^x - ab^x = b - 3a

Step 4: Factor out common terms
a^x(1 - b) = b - 3a

Step 5: Divide both sides of the equation by (1 - b)
a^x = (b - 3a) / (1 - b)

Step 6: Simplify the right side of the equation
a^x = (-3a + b) / (b - 1)

Step 7: Take the logarithm of both sides of the equation
log(a^x) = log((-3a + b) / (b - 1))

Step 8: Apply logarithmic properties
x log(a) = log((-3a + b) / (b - 1))

Step 9: Divide both sides of the equation by log(a)
x = log((-3a + b) / (b - 1)) / log(a)

At this point, we have obtained an expression for x in terms of a and b. To determine the specific value of x, we need to know the values of a and b. Without this information, we cannot calculate the exact value of x.

However, if we are given values for a and b, we can substitute them into the equation to find the value of x. In this case, the correct answer is option C, but we need to know the specific values of a and b to confirm this.