A reliable method
Translate a logarithmic equation to an exponential one. Check the argument is positive before simplifying or solving. Apply product and quotient rules only under their domain assumptions.
The common trap
A logarithm of a sum does not split into a sum of logarithms. Logarithmic arguments cannot be zero or negative in real-valued work.
Make the connection
Exponential and logarithmic models reverse the roles of input and output. An algebraic solution must still satisfy the original domain.
A logarithm asks for an exponent
log₂32=5 because 2⁵=32. Compare log₂(8×4)=5 with log₂8+log₂4=3+2=5: the product rule works. In contrast log₂(8+8)=4 while log₂8+log₂8=6. The addition inside the argument changes the problem and cannot be split. For log₂(x−3)=2, convert to x−3=4, so x=7, then check the argument is positive. A proposed algebraic solution that makes an original logarithm argument zero or negative is invalid in real-number work. Use the exponential translation as an independent verification of the final result.
Transfer, explain and check
Solve log₃(x−1)=2. Check: x=10, and its argument is 9>0. Then explain why x=−8 would fail the domain even before evaluating a logarithm.
What is log₂(8)?
- The logarithm asks for an exponent on 2.
- Check 2^3=8.
- The exponent, hence the logarithm, is 3.
Compare the other answer choices
Divide the argument by the base rather than taking its logarithm.
Multiply the exponent by the base after identifying it.
Report the logarithm argument rather than its exponent.
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 and syllabus