Introduction
Most calculators only have two log buttons: LOG and LN. Understanding these specific bases is essential for calculating pH, sound intensity, and continuous growth.
Prerequisite Connection
You should recall that base is the limit of continuous growth. This naturally leads to the natural log.
Today's Increment
We replace with and with . It's just a notation change, but a crucial one.
Why This Matters
is the "natural" choice because it has the simplest derivative: . Any other base introduces a messy constant factor (), so 99% of calculus is done with natural logs.
Key Concepts
The Common Logarithm
Base 10. Used in engineering (decibels) and chemistry (pH).
If no base is written, assume base 10.
The Natural Logarithm
Base . Used in calculus and continuous growth models.
stands for logarithmus naturalis.
Comparison Graph
Worked Examples
Example 1: Evaluating (Basic)
Evaluate .
Use Definition
asks "to what power must we raise to get ?"
Answer
Example 2: Evaluating (Intermediate)
Evaluate .
Evaluate First Term
.
Evaluate Second Term
.
Sum
Example 3: Solving Equations (Advanced)
Find given .
Convert to exponential
.
Solve
Common Pitfalls
Misreading Notation
does NOT mean . It is a function, like . You cannot "divide by ln" to solve an equation.
Real-World Application
Chemical pH
The acidity of a solution is defined by . A pH of 2 has times more hydrogen ions than a pH of 4. This is why small changes in pH can be deadly for biology.
Practice Quiz
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