Base Breakers Teacher Guide Base Breakers
Teacher Guide
Duration
45 Minutes
Grade Level
High School Algebra 2
Topic
Logarithms (Change of Base)
Lesson Objectives
Explain why the Change of Base formula is necessary for modern computing devices.
Apply the Change of Base formula to evaluate logarithms with non-standard bases.
Compare results between Common Log (base 10) and Natural Log (base \( e \)) to verify formula consistency.
Instructional Flow
5 min
Warm-up: The Exponential Wall
Students mentally solve simple logs. Introduce \( 2^x = 15 \).
Key Discussion Point: "We know \( 2^3=8 \) and \( 2^4=16 \), so \( x \) must be close to 4. But how do we find it exactly?"
5 min
Video Discovery
Watch "Change of Base Formula - Logarithms".
Pause at 0:24: Students copy the formula into their worksheet notes.Pause at 0:57: Check for understanding of base 10 convention. Ask: "Why is no base written on the calculator button?"Pause at 1:28: Have students attempt the two practice problems before revealing answers.
10 min
Calculated Constraints
Discuss the limitation of standard calculators.
Facilitation Tip: Ask students to look at their physical calculators. "Who has a button that lets you type in any base?" Most scientific calculators only have LOG and LN. Explain that the formula is the "bridge" between the math and the hardware.
20 min
Activity: Calculator Relay
Students work in pairs to evaluate a list of 20 logarithms.
Partner A (The Converter)
Writes the log as a fraction (e.g., \( \frac{\log 125}{\log 5} \)).
Partner B (The Operator)
Inputs the fraction into the calculator and records the result.
Partners switch roles every 5 problems.
5 min
Closure: Exit Ticket
Students explain the relationship between \( \log \) and \( \ln \) within the formula.
Prep Checklist
Printables
Student Worksheet (1 per student)
Calculator Guide (1 per pair)
Exit Tickets (pre-cut)
Tech & Tools
Projector & YouTube Access
Scientific Calculators (1 per pair)
Base Breakers Slides Algebra 2
BASE BREAKERS
Unlocking Logs with Change of Base
The Power Wall
Mental Math
\( 2^x = 16 \)
Mental Math
\( 3^x = 243 \)
Wait... what about this?
\( 2^x = 15 \)
Where does the "mental math" break down?
Discovery Clip
Pause 1: 0:24
Pause 2: 0:57
Embedded media
Change of Base Formula
\( \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \)
Common Shortcut
\( \frac{\log b}{\log a} \)
Base 10 is default
Natural Shortcut
\( \frac{\ln b}{\ln a} \)
Base \( e \) works too!
Hardware Limits
Most calculators are built with only two specific logarithm tools:
LOG
Common Log (Base 10)
LN
Natural Log (Base \( e \))
ACTIVITY: CALCULATOR RELAY
PARTNERS
Work in pairs of two. Open your worksheet.
ROLES
Partner A converts.
THE SWITCH
Switch roles every 5 problems!
Goal: Complete all 20 problems as a team!
EXIT TICKET
Explain in one sentence why log₂ 8 can be written as ln 8 / ln 2.
Does it yield the same result?
Base Breakers Worksheet Base Breakers
Change of Base Formula & Application
Name:
Date:
Part 1: The Exponential Wall
Mental Check 1
\( 2^x = 16 \)
Mental Check 2
\( 3^x = 243 \)
The Challenge
\( 2^x = 15 \)
Why is the last problem harder to solve mentally?
Part 2: The Formula
Write the formula as seen in the video
\( \log_a(b) = \quad \quad \quad \quad \quad \)
Part 3: Calculator Relay
Switch roles every 5 problems!
# Problem Fraction Form Result 1 \( \log_2(15) \) 2 \( \log_5(120) \) 3 \( \log_3(40) \) 4 \( \log_8(500) \) 5 \( \log_4(1024) \) 6 \( \log_{1.5}(10) \) 7 \( \log_7(49.5) \) 8 \( \log_{12}(144) \) 9 \( \log_6(200) \) 10 \( \log_3(1000) \)
# Problem Fraction Form Result 11 \( \log_{20}(400) \) 12 \( \log_5(0.2) \) 13 \( \log_9(3) \) 14 \( \log_2(0.5) \) 15 \( \log_8(2) \) 16 \( \log_4(0.125) \) 17 \( \log_{11}(121) \) 18 \( \log_3(81) \) 19 \( \log_7(2401) \) 20 \( \log_2(1024) \)
Scientific Calculator Guide Hardware Manual
Scientific Calculator Reference: Change of Base
3.9069
LOG
LN
LOG BUTTON
LN BUTTON
The Rule
Most calculators only have Common Log (Base 10) and Natural Log (Base \( e \)). To calculate any other base, you must use a fraction.
The Keystrokes
Example: Evaluate \( \log_2(15) \)
1
LOG
2
15
3
)
Very important!
4
/
5
LOG
6
2
7
)
8
ENTER
Error Log: Common Pitfalls
1. Forgetting Parentheses
Most calculators open a parenthesis automatically when you hit LOG. You MUST close it before hitting the division key, or the calculator will think the division is part of the first log!
BAD: LOG( 15 / LOG( 2 ) )
GOOD: LOG( 15 ) / LOG( 2 )
2. Confusing LOG and LN
Good news! For the Change of Base formula, it doesn't matter which button you use—as long as you use the SAME one for both the top and bottom.
MATCH: LN( 15 ) / LN( 2 ) = CORRECT
MIXED: LOG( 15 ) / LN( 2 ) = WRONG
Base Breakers Exit Ticket Exit Ticket
Name:
In one sentence, explain why log₂ 8 can be written as ln 8 / ln 2.
Does it yield the same result as log₂ 8?
Yes
No
Exit Ticket
Name:
In one sentence, explain why log₂ 8 can be written as ln 8 / ln 2.
Does it yield the same result as log₂ 8?
Yes
No
Exit Ticket
Name:
In one sentence, explain why log₂ 8 can be written as ln 8 / ln 2.
Does it yield the same result as log₂ 8?
Yes
No
Exit Ticket
Name:
In one sentence, explain why log₂ 8 can be written as ln 8 / ln 2.
Does it yield the same result as log₂ 8?
Yes
No
