Session 04-03 - Exponential & Logarithmic Functions

Section 04: Advanced Functions

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Review from Session 04-02

Work individually for 5 minutes, then we discuss

Simplify: \(\sqrt[3]{8x^6}\)
What is the domain of \(f(x) = \sqrt{x - 4}\)?
Compare the growth rates: Which grows faster for large \(x\): \(x^3\) or \(x^{3.1}\)?

Homework Discussion - 15 Minutes

Your questions from Tasks 04-02

Focus on power functions and economic applications

Challenges with fractional and negative exponents
Domain determination strategies
Growth rate comparison problems

Exponential functions will show dramatically different growth behavior!

Learning Objectives

Today’s Goals

By the end of this session, you will be able to:

Understand exponential functions and their unique properties
Distinguish between exponential growth and decay
Work with the natural exponential function \(e^x\)
Apply compound interest formulas in finance
Model real-world growth and decay processes
Compare exponential vs. polynomial growth rates
Analyze logarithmic functions, their graphs and transformations
Read semi-log and log-log plots for business data

Introduction to Exponential Functions

From Powers to Exponentials

A fundamental shift in perspective

Power Functions: \(f(x) = x^n\) → variable base, fixed exponent

Exponential Functions: \(f(x) = a^x\) → fixed base, variable exponent

Power: \(x^2\)

Input: \(2\) → Output: \(4\)
Input: \(3\) → Output: \(9\)
Input: \(4\) → Output: \(16\)

Exponential: \(2^x\)

Input: \(2\) → Output: \(4\)
Input: \(3\) → Output: \(8\)
Input: \(4\) → Output: \(16\)

Key Difference: Exponentials grow MUCH faster than any polynomial!

Definition and Properties

The exponential function family

An exponential function has the form: \[f(x) = a \cdot b^x\]

\(a \neq 0\) is the initial value (vertical stretch/reflection)
\(b > 0, b \neq 1\) is the base

Essential Properties:

Domain: All real numbers, Range: \((0, \infty)\) when \(a > 0\)
Always passes through \((0, a)\) and has an inverse

Graphical Behavior

Growth vs. Decay patterns

Exponential or Not?

2 minutes individual, 3 minutes pairs, 2 minutes class discussion

Which of these are exponential functions?

\(f(x) = 5 \cdot 2^x\)
\(g(x) = x^5\)
\(h(x) = 3^{2x}\)
\(k(x) = (-2)^x\)

\(m(x) = e^{-x}\)
\(n(x) = 2^x + 3^x\)
\(p(x) = \pi^x\)
\(q(x) = 1^x\)

Discuss: What makes a function exponential? What are the restrictions?

Exponential Growth Models

The Natural Base e

The most important number in continuous growth

The number \(e ≈ 2.71828...\) is called Euler’s number

Arises naturally in countless real-world processes
Foundation for calculus and differential equations
Base of the natural exponential function \(f(x) = e^x\)

\[e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n\]

Properties of \(e^x\)

What makes e special

The function \(f(x) = e^x\) has unique mathematical properties:

Self-similarity: Rate of change equals value (derivative equals itself)
Series representation: \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...\)
Natural growth: Models any continuous growth or decay process
Perfect inverse: Has as inverse function the natural logarithm

Whenever you see continuous processes in nature, business, or science, \(e\) appears!

Visualizing \(e^x\)

Always positive, passes through \((0, 1)\), grows faster as \(x\) increases

General Growth Formula

Two equivalent ways to model growth

The general exponential growth model can be written as:

Discrete form: \(A(t) = A_0 \cdot b^t\) where \(b > 1\)

Continuous form: \(A(t) = A_0 \cdot e^{kt}\) where \(k > 0\)

\(A_0\) = initial amount
\(b\) = growth factor (how much it multiplies per period)
\(k\) = continuous growth rate
\(t\) = time

These forms are equivalent! Relationship: \(b = e^k\) or \(k = \ln(b)\)

Business Application: Market Growth

User adoption example

A new app has 1,000 users and grows by 30% monthly.

Discrete model: \(U(t) = 1000 \cdot 1.3^t\) where \(t\) is months

Continuous model: \(U(t) = 1000 \cdot e^{0.2624t}\) (since \(\ln(1.3) ≈ 0.2624\))

Calculations:

After 6 months: \(U(6) = 1000 \cdot 1.3^6 ≈ 4,826\) users
After 12 months: \(U(12) = 1000 \cdot 1.3^{12} ≈ 23,298\) users

Question: Why might the continuous form be more realistic here?

Break - 10 Minutes

Exponential Decay Models

Decay Processes

When things decrease exponentially

The exponential decay model:

\(A(t) = A_0 \cdot b^t\) where \(0 < b < 1\)

or: \(A(t) = A_0 \cdot e^{-kt}\) where \(k > 0\)

Question: Any idea where we find this in the real world?

Radioactive decay
Drug concentration in body
Depreciation of assets
Cooling processes

Business Example: Depreciation

A machine costs €50,000 and depreciates 15% annually.

Model: \(V(t) = 50000 \cdot 0.85^t\)

Compound Interest

From Simple to Compound Interest

The power of reinvesting earnings

Simple interest: Only the principal earns interest \[A = P(1 + rt)\]

Compound interest: Interest earns interest \[A = P\left(1 + \frac{r}{n}\right)^{nt}\]

\(P\) = principal (initial investment), \(r\) = annual interest rate
\(n\) = compounding frequency per year, \(t\) = time in years

Discovering e Through Compounding

Let’s compound €1 at 100% interest for 1 year (\(P=1, r=1, t=1\)):

\[A = \left(1 + \frac{1}{n}\right)^n\]

Annually: \((1 + 1)^1 = 2.000\)
Semi: \((1 + 0.5)^2 = 2.250\)
Quarterly: \((1 + 0.25)^4 = 2.441\)
Monthly: \((1.0833...)^{12} = 2.613\)

Daily: \((1.00274...)^{365} = 2.7146\)
Hourly: \(≈ 2.7181\)
Every second: \(≈ 2.71827\)
As \(n \to \infty\): \(e ≈ 2.71828...\)

This limit gives us Euler’s number \(e\), the foundation of continuous growth!

Continuous Compounding

The mathematical limit of frequent compounding

As compounding becomes instantaneous (\(n \to \infty\)): \[A = P \cdot e^{rt}\]

When to use continuous compounding:

Theoretical maximum return calculations
Natural growth processes (populations, investments, …)

Continuous vs. daily compounding differs by under 0.01% in practice. Compounding frequencies and the Effective Annual Rate follow in Session 08-01.

Calculator: Exponential Calculations

Key buttons for exponential functions:

Operation	Keys
\(e^x\)	SHIFT + ln
\(10^x\)	SHIFT + log
\(x^n\)	Use x^ key

Examples:

\(e^2 = 7.389...\)
\(10^{1.5} = 31.62...\)
\(2^{10} = 1024\)

Spot the Error

Can you identify the errors? Work with your neighbor

Time allocation: 5 minutes to find errors, 5 minutes to discuss

Student work:

“Since \(2^3 = 8\), then \(2^{3x} = 8x\)”
“The function \(f(x) = -2^x\) represents exponential decay”
“If inflation is 3% annually, prices double in \(\frac{100}{3} ≈ 33\) years”
“\((e^2)^3 = e^5\)”

Comparing Growth Rates

Exponential vs. Polynomial

Which wins in the long run?

Comparing Different Bases

How base affects growth rate

For exponential growth (\(b > 1\)):

Larger base → faster growth
\(e^x\) grows faster than \(2^x\) but slower than \(3^x\)

For exponential decay (\(0 < b < 1\)):

Smaller base → faster decay
\((0.3)^x\) decays faster than \((0.5)^x\)

Remember: The base determines the rate of growth/decay!

The Power of Exponentials

Small changes, dramatic effects

A cent doubled daily for 30 days:

Day 1: €0.01
Day 10: €5.12
Day 20: €5,243
Day 30: €5,368,709

“The greatest shortcoming of the human race is our inability to understand the exponential function.” - Albert Bartlett

Guided Practice - 25 Minutes

Individual Exercise Block I

Work alone for 5 minutes, then discuss for 5 minutes

Problem 1: Population Growth

A bacteria colony starts with 100 cells and triples every 4 hours.

Write the exponential model \(N(t)\) where \(t\) is hours
How many bacteria after 12 hours?
How many bacteria after 16 hours?
When will the population reach 100,000 cells? (Express as an equation!)

Individual Exercise Block II

Work alone for 5 minutes, then discuss for 5 minutes

Problem 2: Investment Comparison

You have €5,000 to invest for 8 years. Compare:

Option A: 6% annual rate compounded quarterly
Option B: 5.9% annual rate compounded continuously

Calculate the final amount for each option
Which option is better and by how much?
What would the continuous rate need to be to match Option A?

Individual Exercise Block III

Work alone for 5 minutes, then discuss for 5 minutes

Problem 3: Half-Life & Medication

A medication has a half-life of 6 hours. You take 200mg.

Write the decay model \(A(t)\) where \(t\) is hours
How much remains after 12 hours?
How much remains after 24 hours?
After how many half-lives will less than 10mg remain?

Coffee Break - 15 Minutes

Quick Challenge Question

An investment grows from €1,000 to €1,500 in 5 years.

Question: What was the annual growth rate if compounded continuously?

Real-World Applications

COVID-19 Spread Model

Early pandemic growth (before interventions, approximation):

Cases doubled every 3 days, starting from 100 cases

Model: \(C(t) = 100 \cdot 2^{t/3}\)

Without intervention:

Day 9: \(100 \cdot 2^3 = 800\) cases
Day 18: \(100 \cdot 2^6 = 6,400\) cases
Day 30: \(100 \cdot 2^{10} = 102,400\) cases

This demonstrates why an early intervention is crucial and “Flattening the curve” was essential as exponential growth is deceptive initially!

Moore’s Law

Technology advancement

“Computing power doubles every 2 years”

If a processor has 1 billion transistors today:

\[T(t) = 10^9 \cdot 2^{t/2}\]

Predictions:

In 10 years: \(10^9 \cdot 2^5 = 32\) billion transistors
In 20 years: \(10^9 \cdot 2^{10} = 1.024\) trillion transistors

This exponential growth has driven the smartphone revolution and AI advancement.

Special Topics

The Rule of 70

Quick doubling time estimation

For growth rate \(r\)% per period: \[\text{Doubling time} ≈ \frac{70}{r}\]

7% growth → doubles in \(\frac{70}{7} = 10\) periods
2% inflation → prices double in \(\frac{70}{2} = 35\) years
10% return → investment doubles in \(\frac{70}{10} = 7\) years

Why 70? It’s a mathematical approximation that works remarkably well for small growth rates!

Logarithmic Scales

When exponentials look linear (be careful!)

Logarithmic Functions

Logarithmic Functions as Inverses

The logarithm undoes the exponential

\[y = \log_b(x) \quad \Longleftrightarrow \quad b^y = x\]

Domain: \((0, \infty)\) - you can only take logs of positive numbers
Range: all real numbers
The graph of \(\log_b(x)\) is the mirror image of \(b^x\) across \(y = x\)
Passes through \((1, 0)\) since \(\log_b(1) = 0\)

The logarithm laws themselves were covered in Session 01-05 - today we study the function and its graph.

Logarithmic Transformations

Spot the Error: Logarithm Mistakes

Find and fix the errors!

Problem: Solve \(\log_2(x) + \log_2(x - 2) = 3\)

Student Solution:

“\(\log_2(x) + \log_2(x - 2) = 3\)

\(\log_2(x + x - 2) = 3\)

\(\log_2(2x - 2) = 3\)

\(2x - 2 = 8\)

\(x = 5\)”

Semi-log and Log-log Plots

Semi-log Plot (y-axis log)

When data spans several orders of magnitude:

Exponential growth/decay patterns
Compound interest, population growth

Recognition:

Exponential functions appear as straight lines
Slope represents growth rate

Log-log Plot (both axes log)

When working with power law relationships:

Power law relationships
Allometric scaling
Economic relationships (supply/demand curves)

Recognition:

Power functions \(y = ax^b\) appear as straight lines
Slope equals the exponent \(b\)

Let’s viusalize both!

Business Application: Market Analysis

Power Law Example: Production Costs

On the log-log plot, the straight line confirms a power law relationship. The slope of -0.3 means that doubling production reduces per-unit cost by about 19% (2^(-0.3) ≈ 0.81).

Mixed Growth Models

Logistic Growth

When exponential growth has limits

Real populations can’t grow forever. The logistic model:

\[P(t) = \frac{L}{1 + Ae^{-kt}}\]

\(L\) = carrying capacity (maximum sustainable population)
\(A\) = related to initial population: \(P(0) = \frac{L}{1+A}\)
\(k\) = growth rate (how fast it approaches capacity)
\(t\) = time

Three Phases of Logistic Growth

Phase 1: Slow Start (Lag Phase)

Population small, resources abundant
Growth rate increasing but total growth still slow

Phase 2: Rapid Growth (Exponential-like Phase)

Population reaches \(\frac{L}{2}\) (inflection point)
Maximum growth rate occurs here

Phase 3: Saturation (Plateau Phase)

Approaching carrying capacity \(L\) as resources become scarce

Logistic Growth Visualization

Real-World Logistic Examples

Where you encounter S-curves in practice

Business & Technology:

Product adoption: Smartphones, social media platforms
Market penetration: New products reaching saturation

Biology & Social:

Population ecology: Species limited by food/space
Rumor/news spreading: Reaches everyone who will hear it

Unlike pure exponential growth (which is unsustainable), logistic growth is realistic. Every real system has limits!

Practice: Logistic Growth Application

Work individually for 5 minutes, then discuss

A new social media platform launches with 100 users. The market can support a maximum of 10,000 users (carrying capacity). The growth follows a logistic model where \(t\) is in months.

\[P(t) = \frac{10000}{1 + 99e^{-0.5t}}\]

Verify that \(P(0) = 100\) (initial population)
Calculate the population after 6 months
After 12 months? What happens as \(t \to \infty\)?

Comparing Growth Models

Exponential vs. Logistic - A visual comparison

Synthesis & Applications

Real-World Application: pH Scale

Where logarithmic properties are important:

\[\text{pH} = -\log_{10}[\text{H}^+]\]

where \([\text{H}^+]\) is hydrogen ion concentration in mol/L

Domain: \((0, \infty)\) for concentration
Range: Typically 0-14 for pH
Each unit change = 10× concentration change
pH 7 is neutral (\([\text{H}^+] = 10^{-7}\))

Orange Juice

If orange juice has pH = 3.5:

Question: How much more acidic is orange juice compared to neutral water?

\[3.5 = -\log_{10}[\text{H}^+]\] \[[\text{H}^+] = 10^{-3.5} \approx 3.16 \times 10^{-4} \text{ mol/L}\]

This is 1000 times more acidic than neutral water!

Profit with Components I

A firm’s profit depends on portfolio size \(x\) (number of clients):

\[P(x) = \frac{80x}{x + 5} - 15\ln(x + 1) + 10\]

Rational term \(\frac{80x}{x+5}\): Revenue per client (approaches 80k asymptote)
Logarithmic term \(-15\ln(x+1)\): Overhead & complexity costs
Fixed term \(+10\): Base profit offset
Key insight: Logarithmic costs eventually erode the revenue gains
Critical question: What’s the optimal client portfolio size?

Profit with Components II

Initially, revenue growth outpaces cost growth → profits increase. Eventually, costs catch up and overtake revenue → profits decline!

Wrap-Up & Key Takeaways

Key Takeaways

Today’s essential concepts

Exponentials grow faster than any polynomial eventually
The base determines the rate of growth or decay
Natural exponential \(e\) appears in continuous processes
Compound interest shows the power of exponential growth
Rule of 70 provides quick intuition for doubling/halving

Final Assessment

5 minutes - Individual work

A new technology startup’s user base is growing exponentially. They started with 1,000 users and now have 4,000 users after 2 years.

Write the exponential growth model \(N(t) = N_0 \cdot b^t\) (find \(b\))
How many users will they have after 5 years?
Is this discrete or continuous growth? What would the continuous model be?
Using the Rule of 70, approximately when will their user base double from the current 4,000?

Next Session Preview

Session 04-04: Introduction to Trigonometric Functions

Modeling periodic phenomena

Angles in degrees and radians
The unit circle
Sine, cosine, and tangent functions
Amplitude, period, and transformations
Seasonal business cycles and periodic models

Complete Tasks 04-03!

Session 04-03 - Exponential & Logarithmic Functions

Entry Quiz - 10 Minutes

Review from Session 04-02

Homework Discussion - 15 Minutes

Your questions from Tasks 04-02

Learning Objectives

Today’s Goals

Introduction to Exponential Functions

From Powers to Exponentials

Power: \(x^2\)

Exponential: \(2^x\)

Definition and Properties

Graphical Behavior

Think-Pair-Share - 7 Minutes

Exponential or Not?

Exponential Growth Models

The Natural Base e

Properties of \(e^x\)

Visualizing \(e^x\)

General Growth Formula

Business Application: Market Growth

Break - 10 Minutes

Exponential Decay Models

Decay Processes

Business Example: Depreciation

Compound Interest

From Simple to Compound Interest

Discovering e Through Compounding

Continuous Compounding

Calculator: Exponential Calculations

Spot the Error

Comparing Growth Rates

Exponential vs. Polynomial

Comparing Different Bases

The Power of Exponentials

Guided Practice - 25 Minutes

Individual Exercise Block I

Individual Exercise Block II

Individual Exercise Block III

Coffee Break - 15 Minutes

Quick Challenge Question

Real-World Applications

COVID-19 Spread Model

Moore’s Law

Special Topics

The Rule of 70

Logarithmic Scales

Logarithmic Functions

Logarithmic Functions as Inverses

Logarithmic Transformations

Spot the Error: Logarithm Mistakes

Semi-log and Log-log Plots

Semi-log Plot (y-axis log)

Log-log Plot (both axes log)

Business Application: Market Analysis

Power Law Example: Production Costs

Mixed Growth Models

Logistic Growth

Three Phases of Logistic Growth

Logistic Growth Visualization

Real-World Logistic Examples

Practice: Logistic Growth Application

Comparing Growth Models

Synthesis & Applications

Real-World Application: pH Scale

Orange Juice

Profit with Components I

Profit with Components II

Wrap-Up & Key Takeaways

Key Takeaways

Final Assessment

Next Session Preview