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[See also Common Logarithms]
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| Natural Logarithm (ln x = logex)
e=2.718281828459... |
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What is a Natural Logarithm?
A natural logarithm is a logarithm with base e. The number e (approximately 2.718281828459) is a special mathematical constant. The natural logarithm is often written as "ln".
Breaking it Down
- Logarithm Definition:
- A logarithm answers the question: "To what power must we raise a certain number (called the base) to get another number?"
- For natural logarithms, the base is e.
- Natural Logarithm:
- The common logarithm uses e as the base.
- Example: ln e2 = 2 because e2 = e2.
Why is it Useful?
- Growth and Decay: Natural logarithms are used in natural growth and decay processes, like population growth, radioactive decay, and compound interest.
- Calculus: They play a crucial role in calculus, especially in dealing with integrals and derivatives.
- Complex Calculations: They simplify complex mathematical models and equations in various fields.
Intuition
Think of natural logarithms as a way of measuring how many times you multiply e to get a number.
- For growth: If a population grows continuously at a rate proportional to its size, the natural logarithm can describe the time it takes to reach a certain size.
- For decay: If a substance decays at a rate proportional to its current amount, the natural logarithm can describe the time it takes to decay to a certain amount.
Key Points to Remember
- Natural logarithms have a base of e.
- ln x is the natural logarithm of x.
- It tells you the exponent needed for e to become x.
Some Properties of Natural Logarithms
ln 1 = 0
ln e = 1
ln xy = (ln x) + (ln y)
ln x/y = (ln x) - (ln y)
ln xn = n(ln x)
eln x = x
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