Common Mistakes

Frequent Errors

1. Accepting $x$ that makes a log argument $\le 0$. 2. Solving $b^x=c$ as $x=c/b$. 3. Dropping absolute-value / Dom awareness after exponentiating. 4. Reversing inequality incorrectly (or failing to reverse when needed). 5. Inverse steps out of order for $ab^{x-h}+k$. 6. Forgetting the inner shift $(x-h)$ in log inverses. 7. Using properties that change Dom without rechecking. 8. Mixing $\ln$ and $\log_{10}$ mid-solution without conversion.

Why These Happen

Students race to an algebraic answer and skip the Dom/context filter that 2.13.A.2 requires.

How to Fix Them

• Two-column work: Solve | Check.
• Number-line test points for inequalities.
• Write undo order: "subtract $k$, divide by $a$, ..." before algebra.

Avoidance Scripts

• "If it is a log equation, check the argument."
• "Extraneous means algebraically found but invalid."
• "Inverse undoes in reverse order."

Quick Self-Check

• Did every solution pass Dom?
• Is inequality direction justified?
• Does composing $f$ and $f^{-1}$ return $x$?