CThere is no sum rule for logarithms; log(5 + 3) = log(8), which cannot be further simplified using log laws
DThe student should have written log(5 + 3) = log(5) − log(3)
The logarithm product rule converts multiplication into addition: log(xy) = log(x) + log(y). There is no analogous rule for sums — log(a + b) does not simplify further using log laws and is not equal to log(a) + log(b). This is one of the most common student errors. The correct evaluation is simply log(8). Confusing the product rule with a nonexistent sum rule likely comes from over-generalizing the product → addition pattern.
Question 2 Multiple Choice
To solve 5^x = 12 for x, which approach is correct?
Ax = 12/5
Bx = log(12) · log(5)
Cx = ln(12) / ln(5), using the change-of-base formula after taking logarithms of both sides
Dx = ln(12) − ln(5)
Taking ln of both sides gives ln(5^x) = ln(12), then using the power rule: x · ln(5) = ln(12), so x = ln(12)/ln(5). This is exactly the change-of-base formula. Option D (ln(12) − ln(5)) would be correct for ln(12/5), not ln(12)/ln(5) — another common confusion between quotient rule and change-of-base. Option A ignores the exponent structure entirely.
Question 3 True / False
The function log_b(x) has domain (0, ∞) and range (−∞, ∞), because it is the inverse of b^x, whose domain is all reals and range is (0, ∞).
TTrue
FFalse
Answer: True
Inverting a function swaps domain and range. Since b^x accepts any real input and always produces a positive output, its inverse log_b(x) must accept only positive inputs and can produce any real output. This is why log(0) and log(negative) are undefined — there is no real exponent that makes b^x equal to 0 or a negative number.
Question 4 True / False
The logarithm law log_b(a + b) = log_b(a) + log_b(b) holds for most positive values of a and b.
TTrue
FFalse
Answer: False
There is no sum rule for logarithms. The product rule log_b(xy) = log_b(x) + log_b(y) converts multiplication into addition, but there is no corresponding identity for sums. For example, log(2 + 3) = log(5) ≈ 0.699, while log(2) + log(3) = log(6) ≈ 0.778 — these are not equal. This is listed as one of the most common misconceptions for this topic.
Question 5 Short Answer
Why does the product rule log_b(xy) = log_b(x) + log_b(y) hold? Explain using what logarithms represent.
Think about your answer, then reveal below.
Model answer: A logarithm is an exponent: log_b(x) = m means b^m = x, and log_b(y) = n means b^n = y. Therefore xy = b^m · b^n = b^(m+n) by the exponent product rule. Taking log_b of both sides: log_b(xy) = m + n = log_b(x) + log_b(y). The product rule works because multiplying two numbers corresponds to adding their exponents — and log_b is asking 'what is the exponent?' So the product of the numbers corresponds exactly to the sum of their logarithms.
This derivation shows that all three log laws (product, quotient, power) are just restatements of exponent rules. If you forget a log law, you can re-derive it immediately from the corresponding exponent rule. This also explains why there is no sum rule: there is no exponent rule of the form b^(m+n) = b^m + b^n — exponents don't add when you add the bases.