Common Mistakes

Frequent Errors

1. Writing the inverse of $b^x$ as $1/b^x$ or $x^b$. 2. Giving Dom$(\log_b x)$ as all reals. 3. Reflecting over the $x$- or $y$-axis instead of $y=x$. 4. Swapping which variable changes multiplicatively. 5. Forgetting $f(g(x))=x$ only on valid domains ($x>0$ for $\log_b$ outer, etc.). 6. Treating $a\log_b x$ as if $a$ changes the base. 7. Using pairs from $ab^x$ without adjusting when initial value is not 1 (topic focuses on initial value 1). 8. Claiming exponential and log graphs look the same without reflection.

Why These Happen

Students blend reciprocal language with inverse language and mix additive/multiplicative stories.

How to Fix Them

• Always swap pairs or reflect over $y=x$.
• Composition check with a concrete number.
• Two sentence templates for growth contrast.
• Domain checklist for logs: argument $>0$.

Avoidance Scripts

• "$b^x$ and $\log_b x$ undo each other."
• "Exponential: add $x$, multiply $y$. Log: multiply $x$, add $y$."
• "Reflect over the diagonal $y=x$."

Quick Self-Check

• Is the inverse a log with the same base?
• Did pairs swap correctly?
• Is growth language matched to the right family?