Algebra II · 35 minutes
Suggested grades 10–12 · Unit 2: Complex solutions and exponential models · Lesson 2 of 2
Exponential Models and a Logarithmic Solution
Solve for an exponent while stating the model’s assumptions.
Mr. S Teaches · Original starter lesson · HS-ALG2-04 · Grade range is planning guidance.Watch, notice, discuss
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Solving Exponential Equations
By The Organic Chemistry Tutor
Compare logarithmic solving steps with the lesson’s exponential model and round only after interpreting units.
Open on YouTube: Solving Exponential Equations (opens a new tab) ↗Solve Exponential Equations With Logarithms: Intro
By Euler's Academy
Use a second explanation of isolating the exponential term, taking a logarithm and checking the solution.
Open on YouTube: Solve Exponential Equations With Logarithms: Intro (opens a new tab) ↗Introduction to Logarithms
By Khan Academy
A conceptual refresher on logarithms as answers to exponent questions before solving the model.
Open on YouTube: Introduction to Logarithms (opens a new tab) ↗How does a logarithm isolate the unknown exponent, and what does that solution mean in context?
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Learning objective
Express a model’s solution using a logarithm and evaluate it appropriately.
Before you begin
- Exponent laws
- Function notation
- Calculator logarithms
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- exponential model
- A rule with the variable in an exponent and constant multiplicative change.
- logarithm
- The exponent required on a specified positive base to produce a value.
- doubling time
- The modeled interval for multiplication by two.
Understand the concept
A fictional archive backup begins at 20 GB and doubles every three months: B(t)=20·2^(t/3). Finding t for a target size reverses exponentiation. A logarithm expresses that exponent exactly, while a calculator gives an approximation. This model assumes the doubling pattern persists; storage capacity or policy may change it.
Teach it step by step
- Calculate B(0), B(3) and B(6) to identify initial size and doubling interval.
- Set 20·2^(t/3)=100 and divide by 20.
- Write t/3=log₂5 and convert to t=3 ln 5/ln 2.
- Evaluate and compare with the 6- and 9-month table values to check reasonableness.
Make the thinking visible
Worked example or model response
t=3log₂5≈6.97 months. The target lies between B(6)=80 and B(9)=160 GB, supporting the interval. If checks occur only at whole months, month 7 is the first listed check at or above 100 GB.
Guided practice
Solve 10·2^(t/2)=80, then interpret t in the model.
Independent practice
1. Solve 5·10^(2t)=500. 2. Solve 4e^t=12 using natural logarithms. 3. State one reason a long-range extrapolation could fail.
Quick assessment
Why can taking a logarithm not produce a real answer for 2^t=−3?
Answer key & assessment criteria
Answers and teaching guidance
- Guided:2^(t/2)=8=2³, so t=6 time units.
- Independent 1:10^(2t)=100, so t=1. 2: t=ln 3≈1.099. 3: A changing growth rate, capacity constraint or policy can invalidate the model.
- Quick check: A positive base raised to a real power is positive; it cannot equal−3.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Build a powers-of-two table before using logarithms and label continuous versus discrete time.
Extension challenge
Compare two backup models with different initial sizes and doubling periods at a stated date.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSF.LE.A.4
Common Core · High school · Functions
Regular high school mathematics standard
Solve exponential models with bases2,10 or e by logarithms and technology.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.