To solve 5^x = 13, a student writes: '5x = 13, so x = 13/5 = 2.6.' What error did this student make?
AThe student should have used natural log instead of common log
BThe student incorrectly treated the exponent as multiplied by the base, skipping the logarithm step
CThe student forgot to verify that 13 is positive before taking a logarithm
DThe equation has no solution because 13 is not a power of 5
The power rule for logarithms states log(b^x) = x·log(b), but this only applies AFTER taking the logarithm of both sides. You cannot bring down the exponent directly. The correct approach: log(5^x) = log(13) → x·log(5) = log(13) → x = log(13)/log(5) ≈ 1.594. The student's answer of 2.6 treats the equation as if it were 5·x = 13, which completely ignores what it means for a variable to be in an exponent.
Question 2 Multiple Choice
Which equation is solved MOST EFFICIENTLY using the common base method (rewriting both sides as powers of the same base)?
A7^x = 50
B3^x = 15
C4^x = 32
D2^x = 10
4^x = 32 can be rewritten as 2^(2x) = 2^5, so 2x = 5 and x = 5/2. Both sides are expressible as powers of 2. The other equations have no convenient common base (7, 50, 3, 15, 10 don't simplify to powers of the same base with small integers), so they require the logarithm strategy. Recognizing when common base is possible saves work and gives exact answers.
Question 3 True / False
You can simplify log(2^x + 5) as x·log(2) + log(5) using logarithm properties.
TTrue
FFalse
Answer: False
Log properties apply to products, quotients, and powers — not sums. log(A + B) ≠ log(A) + log(B), and there is no property that distributes log across addition. The power rule log(A^n) = n·log(A) requires the entire argument to be raised to a power, not added to something else. The expression log(2^x + 5) cannot be simplified with standard log properties, and trying to do so is one of the most common errors when solving exponential equations.
Question 4 True / False
When solving an exponential equation by taking the logarithm of both sides, it does not matter which logarithm base you use — you will get the same numerical answer.
TTrue
FFalse
Answer: True
The change-of-base formula guarantees this: log_a(x)/log_a(b) = log_c(x)/log_c(b) for any valid bases a and c. For example, x = log(20)/log(3) = ln(20)/ln(3) = log₂(20)/log₂(3) ≈ 2.727 in all cases. The choice between log₁₀ and ln is one of computational convenience, not correctness. For equations with base e (like e^x = k), using ln is most convenient since ln(e^x) = x exactly.
Question 5 Short Answer
Explain why taking the logarithm of both sides is the key step that allows you to solve 3^x = 20, and what logarithm property makes this work.
Think about your answer, then reveal below.
Model answer: Taking the logarithm of both sides converts an equation where the variable is trapped in an exponent into one where the variable is a coefficient you can isolate algebraically. The power rule for logarithms — log(b^x) = x·log(b) — moves the exponent down as a multiplier, but only after the log is applied: log(3^x) = log(20) → x·log(3) = log(20) → x = log(20)/log(3). This works because logarithm and exponential are inverse operations — applying log undoes the exponential the same way division undoes multiplication.
The key is sequencing: you cannot apply the power rule to bring down the exponent until after you have taken the logarithm of both sides. Students who skip this step treat 3^x as if it were 3·x. Once the log is applied, the power rule transforms the equation from exponential to linear, and basic algebra handles the rest.