Introduction
A logarithm is simply an exponent in disguise. We learn to translate between exponential and logarithmic forms to solve for unknown powers.
Past Knowledge
You should be comfortable with exponents, specifically that and negative exponents usually mean fractions.
Today's Goal
We are learning the inverse language: asks "Base to what power is ?"
Future Success
Logarithms are the key to integrating functions like . While works for most powers, it fails for . The answer is .
Key Concepts
The Logarithm Definition
For :
Since , we write .
We interpret as "The exponent on that gives ".
Worked Examples
Example 1: Exponential Form (Basic)
Write in exponential form.
Identify components
Base , Exponent , Result .
Rearrange
Example 2: Evaluating Expressions (Intermediate)
Evaluate .
Set equal to unknown variable
means .
Find the power
We know . Since it's a fraction, the exponent must be negative.
Answer
Example 3: Solving for Base (Advanced)
Find if .
Rewrite in exponential form
.
Take the 4th root
.
Answer
Common Pitfalls
Confusing Input and Output
Don't confuse with . The log asks for the exponent required to get 8, which is 3. It's a shrinking function, not a growing one.
Real-World Application
The Richter Scale
Earthquakes are measured logarithmically. A magnitude 7.0 quake is not just "a little" stronger than a 6.0 one—it's (10x) the amplitude. A magnitude 8.0 is (100x) stronger than a 6.0.
Practice Quiz
Practice Quiz
Loading...