How to Solve Logarithm Problems
Solve logarithm aptitude problems using product, quotient and power rules with exponential conversion, a worked example and practice questions.
Expected Interview Answer
A logarithm answers 'to what power must the base be raised to get this number' — log_b(x) = y means b^y = x — and logarithm problems are solved by converting between this exponential and logarithmic form and applying the product, quotient, and power rules.
The three core rules are: log_b(m×n) = log_b(m) + log_b(n) (product becomes sum), log_b(m/n) = log_b(m) − log_b(n) (quotient becomes difference), and log_b(m^k) = k×log_b(m) (a power pulls out as a multiplier). Converting bases uses the change-of-base formula, log_b(x) = log_a(x)/log_a(b), useful when a calculator only provides base-10 or base-e logs. Because a logarithm is just the inverse of exponentiation, every logarithm equation can be rewritten as an exponential equation and solved with index laws, and vice versa. Always check the domain: logarithms are only defined for positive arguments and a positive base not equal to 1.
- Product/quotient/power rules turn multiplication into addition
- Change-of-base lets any log be computed from base-10 or base-e
- Log and exponential forms are interchangeable, doubling your solving options
AI Mentor Explanation
Asking 'how many overs of doubling would take a score of 8 to reach 64' is asking for log base 2 of 8, which equals 3, since 2 cubed is 8... reversed here, the question 'how many doublings from 1 reach 64' is log base 2 of 64 equals 6. This is exactly what a logarithm does: it recovers the exponent when you know the base and the result, the mirror operation to exponentiation. Combining two doubling streaks by multiplying their run totals corresponds to adding their logarithms, the product rule in action.
Worked example
Combine via product rule
- log₂(x(x−2)) = 3
Convert to exponential
- x(x−2) = 2^3 = 8
Solve and check domain
- x = 4 (x = −2 rejected)
Step-by-Step Explanation
Step 1
Recall the definition
log_b(x) = y means b^y = x — convert between forms freely.
Step 2
Apply product/quotient/power rules
Combine or split logs to isolate the unknown.
Step 3
Convert to exponential form
Once isolated, rewrite log_b(x) = y as x = b^y to solve.
Step 4
Check the domain
Reject any solution making a log argument zero or negative.
What Interviewer Expects
- Fluent conversion between logarithmic and exponential form
- Correct use of product, quotient, and power rules
- Awareness of the change-of-base formula
- Domain-checking to reject invalid (non-positive argument) solutions
Common Mistakes
- Treating log(m+n) as log(m) + log(n) — the product rule needs multiplication, not addition
- Forgetting to check that logarithm arguments stay positive
- Misapplying the power rule to the base instead of the argument
- Confusing log base 10 with natural log (base e) without checking the notation
Best Answer (HR Friendly)
“I treat a logarithm as just the inverse of an exponent — log base b of x is the power you raise b to in order to get x. Using the product, quotient, and power rules I combine or split the log terms until I can isolate the unknown, then convert back to exponential form to solve. The last step I always do is check the domain, since a logarithm’s argument has to stay positive.”
Follow-up Questions
- How would you compute log_5(20) using only a calculator with log base 10?
- Why is the logarithm of a negative number undefined for real numbers?
- How do you solve an equation with logs on both sides but different bases?
- What is the relationship between log and exponential graphs?
MCQ Practice
1. log_2(8) = ?
2^3 = 8, so log_2(8) = 3.
2. Simplify: log(20) − log(4)
Quotient rule: log(20/4) = log(5).
3. Solve for x: log_3(x) = 4
Convert to exponential form: x = 3^4 = 81.
Flash Cards
Definition of log_b(x)? — The power y such that b^y = x.
Product rule for logs? — log_b(m×n) = log_b(m) + log_b(n).
Power rule for logs? — log_b(m^k) = k × log_b(m).
Change-of-base formula? — log_b(x) = log_a(x) / log_a(b).