A reliable method
Translate log_b(a) = c into b^c = a. Use that equivalence to solve exact cases before applying calculator approximations or logarithm laws.
The common trap
A logarithm of a sum cannot be split into a sum of logarithms. The product and quotient rules require multiplicative structure and positive arguments.
Check and transfer
After solving a logarithmic equation, check that every original argument is positive. Algebraic manipulation can produce candidates outside the domain.
Worked comparison: identify the structure
The statement log₃(81)=4 is equivalent to 3⁴=81. Dividing 81 by 3 gives 27, which is not the logarithm. In contrast, log₃(1/81)=−4 because a negative exponent creates a reciprocal.
Transfer and explain
If log₂(a)=3 and log₂(b)=5, then log₂(ab)=8, not 15. The product inside the logarithm becomes an addition outside. Check using a=8, b=32 and ab=256=2⁸.
What is log₃(81)?
- A logarithm asks for the exponent on its base.
- Express the argument as an exact power of 3.
- The requested value is 4. Substitute or reverse the operation to check this result.
Compare the other answer choices
Multiply base and exponent rather than report the exponent.
Treat a positive power as a reciprocal power.
Reverse the logarithm base and argument, taking the reciprocal 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.
CLEP College Algebra exam overviewSiyavula Grade 10 foundation reference