The decision and its conditions
Logarithm rules translate products into sums and quotients into differences only for valid positive arguments. The base must be positive and different from one. Change of base is a ratio of logarithms in the same valid new base. Combining logarithms can turn an equation into a quadratic, but each original logarithm must still exist. A negative algebraic candidate may therefore be excluded even when it satisfies the combined polynomial. A variable base also needs its own restrictions; do not solve an exponent relation and forget what makes the original logarithm meaningful.
Worked example
Using base-two logarithms, evaluate log₂(8)/log₂(4) exactly. The numerator is 3 because 2^3=8. The denominator is two. The ratio is 3/2, equivalently log₄(8).
A contrasting decision
Solve log₂(x)−log₂(x−2)=1 over the reals. The original arguments require x>2. The quotient rule gives x/(x−2)=2, so x=2x−4. Thus x=4; both original arguments are positive.
Diagnose a competing method
The first example’s tempting answer 1 follows this mistake: Subtract logarithms instead of dividing. Change of base is a quotient, not a difference, of logarithms.
Transfer before revealing
Solve log₂(x)+log₂(x+3)=log₂(40) over the reals. Pause and identify the original restrictions. Solution: Both logarithm arguments must be positive, hence x>0. The product equation is x(x+3)=40; its roots are 5 and -8. Only x=5 is allowed by the original logarithms.
Check the boundary of the method
A negative algebraic candidate may therefore be excluded even when it satisfies the combined polynomial. Turn the difference of logs into a quotient and retain both positivity conditions. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
Using base-two logarithms, evaluate log₂(8)/log₂(4) exactly.
- The numerator is 3 because 2^3=8.
- The denominator is two.
- The ratio is 3/2, equivalently log₄(8).
Compare the other answer choices
Subtract logarithms instead of dividing.
Ignore the denominator logarithm.
Reverse the change-of-base ratio.
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 overview