Learning goal and prerequisites
Use logarithm rules without retaining forbidden or extraneous candidates. Before starting, review logarithmic functions.
Method and conditions
Every logarithm argument must be positive. Establish these inequalities before combining logarithms. Convert a single log equality to an exponential equality; solve and check the original arguments. Base greater than one preserves inequality direction.
Worked example: show the decisions
Solve log₂(x−2)+log₂(x+2)=5. The domain is x>2. Combining gives log₂(x²−4)=5, hence x²=36 and candidates ±6. Only x=6 survives the original domain.
Contrast: why a plausible shortcut fails
Combining the two logs and checking only x²−4>0 would also admit x<−2; that is not the domain of the original sum. Domain checking belongs before and after manipulation.
Transfer: try before revealing the answer
Count integer x satisfying log₂(x−3)<3. State both the lower and upper boundary conditions.
Transfer solution and check
Positivity gives x>3; monotonicity gives x−3<8, hence x<11. Integers 4 through 10 give seven solutions. Neither boundary is allowed.
Solve log₂(x−3)+log₂(x+3)=log₂(16). Give the valid real x.
- The original logs require x>3; combine into x²−9, then retain the positive admissible root.
- Using these quantities: √(16+9) = 5.
- Check the original domain, requested quantity and units; do not substitute a related parameter.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus Course at a Glance