The Number e and the Natural Logarithm
A free Algebra II lesson from the “Logarithms and the Natural Base” unit, with a worked example and practice problems including step-by-step solutions.
The number e (approximately 2.71828) is the base of the natural exponential function e^x and the natural logarithm ln(x). Because ln is the inverse of e^x, ln(e^a) = a and e^(ln(a)) = a. Continuous compound interest uses A = P * e^(rt), where P is the principal, r is the rate as a decimal, and t is time.
What you'll learn
- Recognize the constant e (approximately 2.718) as the base of natural exponential and natural log
- Use ln(x) as the inverse of e^x: ln(e^a) = a and e^(ln(a)) = a
- Apply the continuous-compounding formula A = P * e^(rt)
Worked example
Problem. Evaluate ln(e^5).
- ln and e^x are inverses: ln(e^a) = a.
- ln(e^5) = 5.
Answer: 5
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Evaluate e^0.
Show solution
- Anything to the 0 power is 1.
Answer: 1
2. Evaluate ln(1).
Show solution
- e^0 = 1, so ln(1) = 0.
Answer: 0
3. Evaluate ln(e).
Show solution
- e^1 = e, so ln(e) = 1.
Answer: 1
4. Evaluate ln(e^7).
Show solution
- ln(e^a) = a.
Answer: 7
5. Evaluate e^(ln(11)).
Show solution
- e^(ln(a)) = a.
Answer: 11
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