Natural logarithm and the number e
The Natural Logarithm and e
The Natural Logarithm and e is a topic in Exponential & Logarithmic Functions in the California Common Core State Standards. It is aligned to Standard F-LE.4, which requires students to express a logarithm as the solution to an exponential model, including the natural base e.
The natural base \(e\approx2.718\) is the base of continuous growth, and the natural log \(\ln\) is the logarithm base \(e\) and the inverse of \(e^x\).
Theory
The natural base \(e\approx2.718\) is the base of continuous growth. Its logarithm is the natural log:
Continuous growth is modeled by \(A=Pe^{rt}\).
Natural log and growth:
How to use e and ln
- Recognize base \(e\) — apply \(\ln\).
- Use \(\ln(e^x)=x\) to unwrap exponents.
- For continuous growth, use \(A=Pe^{rt}\).
- Evaluate \(\ln\) with a calculator.
Use the definition of \(\ln\).
| \(\ln e\) | \(=\) | \(1\) |
| \(\ln 1\) | \(=\) | \(0\) |
Take the natural log of both sides.
| \(x\) | \(=\) | \(\ln 7\) |
| \(\approx\) | \(1.946\) |
Use \(A=Pe^{rt}\).
| \(A\) | \(=\) | \(500e^{0.04(10)}\) |
| \(=\) | \(500e^{0.4}\approx\$745.91\) |
\(\ln\) and \(e^x\) undo each other.
| \(\ln(e^5)\) | \(=\) | \(5\) |
Common pitfalls
Frequently asked questions
What is the number e?
The natural base, \(e\approx2.718\), used for continuous growth.
What is the natural logarithm?
\(\ln x=\log_e x\), the logarithm base \(e\).
What is \(\ln(e^x)\)?
\(x\) — the functions are inverses.
What models continuous growth?
\(A=Pe^{rt}\).