Introduction
Until now, we’ve evaluated functions like by plugging in . But what if we know the result is 8 and want to find ? That’s where logarithms come in.
Prerequisite Connection
You should be comfortable with exponents, specifically that and negative exponents usually mean fractions.
Today's Increment
We are learning the inverse language: asks "Base to what power is ?"
Why This Matters
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
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