Logarithmic Solutions: Real-World Applications

Lesson 2 of 2: Doubling Time, Half-Life, and Mixed Models

In this lesson:

• Apply the three-step procedure to real-world models
• Interpret solutions in context with units
• Recognize doubling time and half-life as special cases

What You Will Be Able to Do

By the end of this lesson, you should be able to:

5. Interpret the solution in context — state what represents, with units and a complete sentence
6. Apply the logarithmic solution method to exponential models involving doubling time, half-life, and target-value problems

Hook: Does the Starting Amount Matter?

An investment grows continuously at 6% per year.

When does it double? — Does the starting amount change the answer?

• Start with $1,000 → doubles to $2,000
• Start with $50,000 → doubles to $100,000

Try to guess: do both scenarios give the same doubling time?

Doubling Time: Why the Initial Value Cancels

Model:

Step 1 — Divide both sides by :

The initial amount cancels completely.

The doubling time is the same regardless of how much you start with

Doubling Time: Completing All Three Steps

From Step 1:

Step 2:

Step 3:

Calculator: , so years

Interpretation: The investment doubles in approximately 11.6 years, regardless of the starting amount.

Half-Life: Setting Up the Model

A radioactive substance has a half-life of 5 years. Starting with 100 grams, when will only 10 grams remain?

Model:

• The base is , written as
• The exponent counts half-life periods
• Goal: find when quantity drops to 10 g

Half-Life Problem: All Three Steps Worked

Step 1 — Divide both sides by 100:

Step 2 — Change-of-base (base 2):

Step 3 — Solve for :

Interpretation: After approximately 16.6 years, only 10 grams remain.

When the Starting Amount Cancels Out

Rate alone determines doubling time and half-life

Check-In: Doubling Time at 8%

Scenario: An account grows continuously at 8% per year.

Set up and solve for the doubling time.

Hint: the setup looks like

What is the doubling time? Compare to the 6% result.

From Special Cases to General Applications

In Chunk 3:

• ✓ Doubling time: initial value cancels
• ✓ Half-life: initial value cancels

In Chunk 4: Mixed real-world problems where you must:

1. Set up the model from a verbal description
2. Apply the three steps
3. Write a complete interpretation sentence

Complete Workflow: Model to Interpretation

• Read: identify , , , and target
• Model: write
• Isolate → Log → Divide: the three steps
• Interpret: answer in a complete sentence with units

Population Growth: When Does a City Quadruple?

Scenario: City of 5,000 grows at 3%/year. When does it reach 20,000?

Step 1:

Step 2: years

Interpretation: The population reaches 20,000 in approximately 47 years.

Interpretation: Saying What the Answer Means

The number answers: when does the population reach 20,000?

Interpretation template:

"After approximately [value] [units], [quantity] reaches [target]."

Written out: After approximately 47 years, the city's population reaches 20,000.

Always identify: what is ? What are the units? What does reaching mean?

Investment Growth: When Does It Quadruple?

Scenario: An account grows continuously at 6%. When does it quadruple?

Model:

Step 1: (P cancels)

Step 2:

Step 3: years

Interpretation: The investment quadruples in approximately 23.1 years.

Radioactive Decay: Solving a Continuous Decay Problem

Scenario: A 200g sample decays at rate . When is only 50g left?

Step 1:

Step 2:

Step 3: years

Interpretation: After approximately 46.2 years, only 50g remains.

Your Turn: Guided Bacteria Growth Problem

Scenario: A colony starts at 300 bacteria and doubles every 3 hours. When does it reach 9,600?

Model:

Step 1 is done:

Complete Steps 2 and 3 — is this a clean logarithm?

Write a complete interpretation sentence for your answer

Independent Practice: Two Mixed Real-World Problems

Solve completely and write an interpretation sentence for each.

1. — when does the quantity reach its target?

2. — when does this quantity triple?

Pause and complete both problems before the next slide

Full Practice Answers with Interpretation Sentences

Problem 1:

After exactly 20 units, the quantity reaches 50,000.

Problem 2:

The quantity triples after approximately 22 time units.

Problem 1: — clean answer, no calculator needed

Complete Five-Step Workflow Reference Chart

Use this as a checklist: one step at a time, in order

Key Takeaways and Misconception Warnings

✓ Isolate → log → divide by — every time
✓ Doubling/half-life: initial value cancels
✓ Interpret with value, units, full sentence

Isolate before taking log
log = base 10, ln = base
Divide by after the log

Coming Up: Logarithmic Functions and Their Graphs

You can now solve for any base

Coming up:

• Graphs of logarithmic functions
• Product, quotient, and power rules
• Exponential and logarithmic functions as inverses