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Algebra 2 · Question of the Day

Algebra 2 Question of the Day

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Friday, October 2, 2026

Use the definition log⁡b(x)=y⇔by=x\log_b(x)=y \Leftrightarrow b^y=x to evaluate ln⁡(e6)\ln(e^6).​

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Question of the Day

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Use the definition log⁡b(x)=y⇔by=x\log_b(x)=y \Leftrightarrow b^y=x to evaluate ln⁡(e6)\ln(e^6).​

  1. e6e^6
  2. 6e6e
  3. 66 (correct answer)
  4. ln⁡(6)\ln(6)

Explanation: This question tests your understanding that logarithms and exponentials are inverse operations—logarithms undo exponentiation and vice versa, just like square roots undo squaring. As inverse operations, logarithms and exponents cancel each other: log_b(bxb^x) = x for any x (the log undoes the exponent), and b^(log_b(x)) = x for x > 0 (the exponent undoes the log). These inverse properties are incredibly useful for simplification: log₃(3⁵) immediately simplifies to 5, and 7^(log₇(20)) immediately simplifies to 20. No calculation needed—they just undo each other! For ln(e⁶), remember that ln is just log_e (natural log with base e). So we're evaluating log_e(e⁶), which by the inverse property log_b(bxb^x) = x simplifies directly to 6. Choice C correctly identifies that ln(e⁶) = 6. Choice A (e⁶) would be the argument itself, not its logarithm. Choice B (6e) incorrectly multiplies 6 and e. Choice D (ln(6)) reverses the problem, taking the natural log of 6 instead of e⁶. The natural logarithm ln and the exponential function with base e are perfect inverses: ln(exe^x) = x and e^(ln(x)) = x. This makes calculations with e and ln particularly clean!