Introduction
We can solve by rooting. But how do we solve ? Rooting won't work because the isn't a power of a number; it is the power.
Prerequisite Connection
You must recall the Power Rule from Lesson 8.5: . This is the mechanism we use.
Today's Increment
We combine isolation strategies (PEMDAS) with the Power Rule to solving for in .
Why This Matters
This technique is strictly required for solving First-Order Differential Equations in Calculus, specifically when modeling Newton's Law of Cooling, population growth, or radioactive decay times.
Key Concepts
The Standard Procedure
- Isolate the exponential term () on one side.
- Take the Natural Log () of both sides.
- Use the Power Rule to bring the exponent down as a coefficient.
- Solve for normally.
Worked Examples
Example 1: The Basics
Solve .
Take Ln of both sides
.
Apply Power Rule
.
Solve
Example 2: Two Steps
Solve .
Isolate Exponential
Subtract 5: .
Divide by 3: .
Log both sides
.
Since , this simplifies to .
Solve
Example 3: Hidden Quadratic (Advanced)
Solve .
Recognize Structure
Let . Then .
Rewrite: .
Factor and Solve for U
.
or .
Back Substitute
Case 1: .
Case 2: . Impossible (exponential cannot be negative).
Common Pitfalls
Taking Logs Too Early
If you have and take logs immediately, you might write . The is not the power of the 2. You MUST divide by 2 first to isolate .
Real-World Application
Forensic Science
A body cools according to Newton's Law of Cooling, which is an exponential decay equation. To determine the time of death, forensic scientists perform the exact algebra we did in Example 2: measure current temp, subtract ambient temp, and solve for using logarithms.
Practice Quiz
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