Base Breakers Worksheet Base Breakers
Inverse Property of Logarithms: \(a^{\log_a x} = x\)
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Date:
Warm-up: Quick Fire
Solve these basic log expressions as fast as you can! (5 minutes)
1. \(\log_2(8) =\)
2. \(\log_5(25) =\)
3. \(\log_{10}(1000) =\)
4. \(\log_3\left(\frac{1}{9}\right) =\)
Video Insights: The Proof
Step 1: The Example Case
How does the video prove that \(7^{\log_7(9)} = 9\)? Sketch the "log of both sides" steps shown:
Step 2: The Base Mismatch
The narrator encounters \(16^{\log_4(x-2)}\). How do they rewrite the 16 to make the property work?
Inverse Property Relay
Use this space to show your work for the Relay Cards. Your team's speed matters, but accuracy is everything!
Round 1
Round 2
Round 3
Round 4
Final Reflection
In exactly ONE sentence, explain why the base of the exponent MUST match the base of the logarithm for the property \(a^{\log_a x} = x\) to hold true.
Base Breakers Slides BASE BREAKERS
Mastering the Inverse Property: \(a^{\log_a x} = x\)
Quick-Fire Warm-up
\[\log_2(8) = ?\]
\[\log_5(25) = ?\]
\[\log_{10}(1000) = ?\]
\[\log_3\left(\frac{1}{9}\right) = ?\]
Clock is ticking... 5 minutes!
The Power of Cancellation
WATCH & LEARN
Embedded media
Focus Point 1
How does base 7 "cancel out" its own logarithm?
Focus Point 2
What happens when the bases don't match?
Pause & Discuss
"How can we make the base 16 match the log base 4?"
16 = 4?
The Golden Rule
\[a^{\log_a(x)} = x\]
The Base Must Match
Example 1
\(5^{\log_5(25)} = 25\)
Example 2
\(10^{\log(x+1)} = x+1\)
Watch Out!
Common Log = Base 10
Inverse Property Relay
Speed meets Precision
The Rules
1 Work in teams of 4.
2 One person solves a card on the whiteboard.
3 Pass the marker to the next teammate.
Goal
Complete all 4 rounds of cards. First team with all correct simplifications wins the "Base Breaker" title!
Reflection
Write ONE sentence on your worksheet explaining why the bases must match for this magic trick to work.
Relay Task Cards Relay Task Cards
Base Breakers Edition
11th Grade Algebra II
Teacher Note: Cut these cards along the dashed lines. Each set (A, B, C, D) represents one round of the relay. Distribute one card at a time to teams as they complete the previous one correctly.
ROUND 1: CARD A
\(6^{\log_6(36)}\)
Simplify completely
ROUND 1: CARD B
\(x^{\log_x(y^2)}\)
Simplify completely
ROUND 2: CARD A
\(10^{\log(x+7)}\)
Simplify completely
ROUND 2: CARD B
\(e^{\ln(5x)}\)
Simplify completely
ROUND 3: CARD A
\(25^{\log_5(3)}\)
Mismatched Bases!
ROUND 3: CARD B
\(9^{\log_3(x+4)}\)
Mismatched Bases!
ROUND 4: CARD A
\(100^{\log(x)} - 4^{\log_2(3)}\)
The Final Challenge
ROUND 4: CARD B
\(8^{\log_2(x-1)} + e^{\ln(4)}\)
The Final Challenge
KEY: A1:36, B1:y^2, A2:x+7, B2:5x A3:9, B3:(x+4)^2, A4:x^2-9, B4:(x-1)^3+4
Base Breakers Teacher Guide Base Breakers
Teacher Implementation Guide
Lesson Duration: 45 Min
Subject: Algebra II
Timeline & Pacing
00:00 - 05:00
Quick-Fire Warm-up
Students solve basic logs on the worksheet. Key: 1) 3, 2) 2, 3) 3, 4) -2.
05:00 - 15:00
Video Viewing & Discussion
Watch the first 2 minutes of the "Exponential Logarithmic Equations" video. Use these pause points:
00:25: Pause. Which terms in the equation look like they could be simplified using a property?01:00: Discuss the proof. Why does "logging both sides" work? (It exposes the exponent as a coefficient).02:10: Pause. How do we make the base 16 match the log base 4? (Rewrite 16 as \(4^2\)).
15:00 - 35:00
Inverse Property Relay
Divide class into teams of 4. Use the Relay Task Cards . One student per round solves a card on the whiteboard, then passes the marker.
Pro Tip: Only give out the "Round 2" card after a team shows you a correct solution for Round 1. This keeps the energy high!
35:00 - 45:00
Reflection & Wrap-up
Students write their 1-sentence reflection. Sample Answer: "The bases must match because the logarithmic and exponential functions only cancel each other out when they are inverses with the same base."
Relay Answer Key
A1: \(6^{\log_6(36)} = 36\)
A2: \(10^{\log(x+7)} = x+7\)
A3: \(25^{\log_5(3)} = (5^2)^{\log_5(3)} = (5^{\log_5(3)})^2 = 3^2 = 9\)
A4: \(100^{\log(x)} - 4^{\log_2(3)} = (10^{\log x})^2 - (2^{\log_2 3})^2 = x^2 - 9\)
B1: \(x^{\log_x(y^2)} = y^2\)
B2: \(e^{\ln(5x)} = 5x\)
B3: \(9^{\log_3(x+4)} = (3^{\log_3(x+4)})^2 = (x+4)^2\)
B4: \(8^{\log_2(x-1)} + e^{\ln(4)} = (2^{\log_2(x-1)})^3 + 4 = (x-1)^3 + 4\)
Common Pitfalls
Students forget that "log" (no base) implies base 10.
Students may try to multiply the bases instead of using exponents when they don't match.
Students may stop at the intermediate step (e.g., leaving it as \((x+4)^2\) instead of recognizing the cancellation happened).
