Log Decoder Slides
Mission: Inverse
Log Decoder
Algebraic methods for inverting logarithmic functions.
Inverse Concept
An inverse function effectively "undoes" the action of the original function.
- Domain of f \(\to\) Range of \(f^{-1}\)
- Switch the x and y values!
Mapping
x
Input
\(f(x)\)
y
Output
The Core Link
Log Form
\[y = \log_{b}(x)\]
Switching
Exp Form
\[b^{y} = x\]
Base Rule: Base b stays the base!
Decode Protocol
01
Set to Y
Replace f(x) with y.
\(y = \dots\)
02
Switch
Swap x and y.
\(x = \log_{b}(y)\)
03
Solve
Convert to Exp form.
\(b^{x} = y\)
04
Label
Use \(f^{-1}(x)\) notation.
\(f^{-1}(x) = \dots\)
Demo Operation
Target
\[f(x) = \log_{4}(x)\]
1
y = \(\log_{4}(x)\)
2
x = \(\log_{4}(y)\)
3
\(4^{x} = y\)
4
\(f^{-1}(x) = 4^{x}\)
Field Test
Alpha
\[g(x) = \log_{2}(x)\]
Decode Area
Beta
\[h(x) = \log_{7}(x)\]
Decode Area
Log Decoder Guided Notes
Log Decoder Notes
Mission: Inverting Logarithmic Functions
Agent:
Date:
1
The Concept
An inverse function effectively _____________________ the original action.
To find it algebraically, we _____________________ the x and y values.
2
The Core Link
Logarithmic Form
\(y = \log_{b}(x)\)
Exponential Form
\(b^{y} = x\)
3
Decode Protocol (Steps)
Step 1:
Replace \(f(x)\) with ________________.
Step 2:
________________ the \(x\) and \(y\) values.
Step 3:
Convert to ________________ form.
Step 4:
Use ________________ notation.
4
Practice Operations
EX 1: \(f(x) = \log_{4}(x)\)
Workspace
EX 2: \(g(x) = \log_{7}(x)\)
Workspace
Extended Field Operations
EX 3: \(h(x) = \log_{2}(x)\) MISSION ID: LVL-01
Analysis Area
EX 4: \(p(x) = \log_{5}(x)\) MISSION ID: LVL-02
Analysis Area
EX 5: \(k(x) = \log_{10}(x)\) MISSION ID: LVL-03
Analysis Area
EX 6: \(q(x) = \log_{0.5}(x)\) Challenge
Analysis Area
Recall: Log base \(b\) always stays the Exponential base \(b\)!
Inverse Mission Assignment
Mission: Inverse
Agent Assignment: Logarithmic Inverses
Agent:
Date:
Protocol
01 \(f(x) \to y\)
02 \(x \leftrightarrow y\)
03 \(b^{x} = y\)
04 \(f^{-1}(x)\)
A
Intelligence Gathering: Conversion
1. \(y = \log_{3}(x)\)
2. \(y = \log_{8}(x)\)
3. \(x = \log_{12}(y)\)
4. \(x = \log_{5}(y)\)
B
Active Operation: Full Decoding
5. \(f(x) = \log_{2}(x)\)
Setup
Solve
6. \(g(x) = \log_{6}(x)\)
Setup
Solve
7. \(h(x) = \log_{1.5}(x)\)
Setup
Solve
Inverse Mission Answer Key
Mission Brief: ANSWERS
Teacher Key: Inverse Mission Assignment
Confidential Key
A
Section A: Conversion
1. \(y = \log_{3}(x)\)
\(3^{y} = x\)
2. \(y = \log_{8}(x)\)
\(8^{y} = x\)
3. \(x = \log_{12}(y)\)
\(12^{x} = y\)
4. \(x = \log_{5}(y)\)
\(5^{x} = y\)
B
Section B: Algebraic Steps
5. \(f(x) = \log_{2}(x)\)
Step 1
\(y = \log_{2}(x)\)
Step 2
\(x = \log_{2}(y)\)
Step 3
\(2^{x} = y\)
Step 4
\(f^{-1}(x) = 2^{x}\)
6. \(g(x) = \log_{6}(x)\)
Step 1
\(y = \log_{6}(x)\)
Step 2
\(x = \log_{6}(y)\)
Step 3
\(6^{x} = y\)
Step 4
\(g^{-1}(x) = 6^{x}\)
7. \(h(x) = \log_{1.5}(x)\)
Step 1
\(y = \log_{1.5}(x)\)
Step 2
\(x = \log_{1.5}(y)\)
Step 3
\(1.5^{x} = y\)
Step 4
\(h^{-1}(x) = 1.5^{x}\)
