To determine the equivalence of the Boolean expression \(AC + B\bar{C}\), we need to employ Boolean algebra rules to simplify and analyze the given options.

Simplification using Boolean Theorems:

1. Given Expression: \(AC + B\bar{C}\)
2. We want to compare it with the options to find an equivalent expression. Let's analyze each option.
3. Since none of the options are directly simple transformations of the given expression, we'll expand and try to simplify it using possible rules.

Examine Each Option:

1. Option 1: \(\bar{A}C + B\bar{C} + AC\)
   • This option cannot be simplified directly to \(AC + B\bar{C}\) as it introduces an additional term \(\bar{A}C\).
2. Option 2: \(\bar{B}C + AC + B\bar{C} + \bar{A}\bar{C}\bar{B}\)
   • This expression is more complex and introduces four terms, which cannot simplify to the original.
3. Option 3: \(AC + B\bar{C} + \bar{B}C + ABC\)
   • Here, the term \(\bar{B}C\) is an extra term, and the expanded form includes additional combinations that do not match \(AC + B\bar{C}\) by direct manipulation.
4. Option 4: \(ABC + \bar{A}B\bar{C} + AB\bar{C} + A\bar{B}C\)
   • Let's analyze this last option in more detail:
   • Break down: \(ABC\) is an intersection of all three inputs.
   • \(\bar{A}B\bar{C}\) captures scenarios where \(\bar{A}\) and \(\bar{C}\) are true.
   • \(AB\bar{C}\) matches part of the \(B\bar{C}\) conjunction.
   • \(A\bar{B}C\) creates another unique true condition.
   • This last option precisely accommodates various scenarios matching the function of \(AC + B\bar{C}\) when expanded and compared under Boolean identities.

Conclusion: After analyzing, the option that matches the original Boolean expression \(AC + B\bar{C}\) in logical equivalence is Option 4: \(ABC + \bar{A}B\bar{C} + AB\bar{C} + A\bar{B}C\). This option expands to handle the truth table equivalent conditions for the given expression.