Session 04-03 - Exponential & Logarithmic Functions

Section 04: Advanced Functions

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

Review from Session 04-02

Work individually for 5 minutes, then we discuss

  1. Simplify: \(\sqrt[3]{8x^6}\)

  2. What is the domain of \(f(x) = \sqrt{x - 4}\)?

  3. Compare the growth rates: Which grows faster for large \(x\): \(x^3\) or \(x^{3.1}\)?

Homework Discussion

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

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

Two panels. Left, exponential growth: 1.5^x, 2^x and 3^x all pass through (0, 1) and rise ever faster to the right, the larger base fastest. Right, exponential decay: 0.3^x, 0.5^x and (2/3)^x all pass through (0, 1) and fall towards zero to the right, the smaller base fastest.

Think-Pair-Share

Exponential or Not?

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

Which of these are exponential functions?

  1. \(f(x) = 5 \cdot 2^x\)
  2. \(g(x) = x^5\)
  3. \(h(x) = 3^{2x}\)
  4. \(k(x) = (-2)^x\)
  1. \(m(x) = e^{-x}\)
  2. \(n(x) = 2^x + 3^x\)
  3. \(p(x) = \pi^x\)
  4. \(q(x) = 1^x\)

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

Exponential Growth

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:

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

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

Visualizing \(e^x\)

Curve of y = e^x for x from -2 to 2. It stays above the x-axis, passes through the marked points (0, 1) and (1, e ≈ 2.718), and gets steeper the further right it goes.

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 \(k = \ln(b)\)

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?

Exponential Decay

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 EUR and depreciates 15% annually.

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

Value of the machine over 10 years. The curve starts at 50,000 EUR and falls steeply at first, then more slowly: about 22,185 EUR after 5 years, and it reaches the dashed 10,000 EUR threshold just before year 10.

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 by Compounding

Compound 1 EUR 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 Functions

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:

  1. “Since \(2^3 = 8\), then \(2^{3x} = 8x\)”

  2. “The function \(f(x) = -2^x\) represents exponential decay”

  3. “If inflation is 3% annually, prices double in \(\frac{100}{3} ≈ 33\) years”

  4. “\((e^2)^3 = e^5\)”

Comparing Growth

Exponential vs. Polynomial

Which wins in the long run?

Two panels comparing x³ and 2^x. From 0 to 5 the polynomial x³ pulls far ahead, reaching 125 against 32. From 0 to 10 the exponential catches up and overtakes just before x = 10, where 2^x is 1024 against 1000 for x³.

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 EUR
  • Day 10: 5.12 EUR
  • Day 20: 5,243 EUR
  • Day 30: 5,368,709 EUR

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

Break

Quick Challenge Question

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

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

Guided Practice

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.

  1. Write the exponential model \(N(t)\) where \(t\) is hours
  2. How many bacteria after 12 hours?
  3. How many bacteria after 16 hours?
  4. 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 EUR to invest for 8 years. Compare:

  • Option A: 6% annual rate compounded quarterly
  • Option B: 5.9% annual rate compounded continuously
  1. Calculate the final amount for each option
  2. Which option is better and by how much?
  3. 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 (every 6 hours, the amount in the body halves). You take 200mg.

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

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 well for small growth rates!

Logarithmic Scales

When exponentials look linear (be careful!)

Two panels showing the same exponential growth, 100 times e^x, for time from 0 to 5. On a linear scale the curve stays low and then shoots upward. On a logarithmic value axis the same data form a straight rising line.

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

Two panels. Left: ln(x) with two transformations; ln(x) + 1 is the same curve shifted up by 1, and ln(x - 1) is shifted right by 1, so it exists only for x > 1. Right: ln(x), the base-10 logarithm and the base-2 logarithm all pass through (1, 0) and exist only for x > 0; the smaller the base, the faster the curve rises.

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 & Log-log

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 visualize both!

Application: Market Analysis

Two panels with the same revenue data, R(t) = 100 e^(0.5t), for years 0 to 5. On a linear scale the six points curve upward ever more steeply, from 100 to about 1218 thousand EUR. On a semi-log scale they lie on a straight rising line, the sign of a constant growth rate, labelled as continuous rate k = 0.5, about 65% per year.

Power Law: Production Costs

Two panels with the same cost-per-unit data, 50 times quantity to the power -0.3, for quantities from 1 to 500. On linear scales the cost drops sharply from 50 and then flattens out. On log-log scales the nine points lie on a straight falling line, the signature of a power law.

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)

Phase 3: Saturation (Plateau Phase)

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

Logistic Growth Visualization

S-shaped logistic growth curve over time from 0 to 10. It starts at 100, rises slowly at first, is steepest where it crosses the dashed line at 500 (the inflection point, half the capacity), and then levels off just below the dashed carrying capacity of 1000.

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

Exponential growth is unsustainable, logistic is realistic. Every real system has limits!

Practice: Logistic Growth

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.

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

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

Comparing Growth Models

Exponential vs. Logistic - A visual comparison

Two growth curves over 24 months, both starting at 100. The dashed exponential curve shoots past the carrying capacity of 10,000 after about 9 months and leaves the chart. The logistic curve follows it at first, then bends into an S-shape and levels off just below the dotted carrying-capacity line.

Synthesis

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 \(10^{3.5} \approx 3{,}162\) 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 (approaches 80k EUR 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

Two panels for P(x) = 80x/(x + 5) - 15 ln(x + 1) + 10 over 0 to 100 clients. Left: the revenue term rises quickly and levels off below its asymptote of 80 thousand EUR, while the logarithmic cost term keeps falling. Right: total profit peaks at about 17 clients with roughly 28.5 thousand EUR and then declines slowly, staying above the break-even line.

Initially, revenue growth outpaces cost growth → profits increase. Beyond the profit peak, costs grow faster than revenue → profits decline!

Coffee Break

FSP Workshop

Problem 1: A Payment App

A start-up runs a payment app for small shops such as cafés and market stalls. For every payment made with the app, the shop pays a fee. In 2026, the app handles 2 million payments, and the fee is 0.15 EUR per payment. The number of payments is expected to grow by 40 % per year. Because of competition, the fee falls by 25 % per year. Let \(t\) be the time in years since 2026.

  1. Give the number of payments \(N(t)\) and the fee \(F(t)\) as exponential functions. Show that the yearly fee revenue is \(R(t) = 300000 \cdot 1.05^t\) EUR. [4 pts.]

Payment App: Growth Rates

  1. A board member says: “The payments grow by 40 % and the fee falls by 25 %, so the revenue grows by 15 % per year.” Assess this statement. Write \(R(t)\) with the continuous growth rate and explain why the continuous rates of \(N\) and \(F\) simply add up. [4 pts.]

Payment App: A Fee Floor

The fee cannot fall below 0.05 EUR, the network costs that the app itself pays for each payment.

  1. Determine the first year in which the fee \(F(t)\) of the model falls below 0.05 EUR, and check this year and the year before. Compute the fee revenue in that year if the app charges 0.05 EUR per payment instead. [5 pts.]

Hint: The larger of two positive numbers has the larger logarithm.

(for verification: \(F(t) = 0.15 \cdot 0.75^t\))

Payment App: Fee Cuts

The app wants its fee revenue never to shrink, even if competition forces larger fee cuts. The payments still grow by 40 % per year.

  1. Determine the largest yearly fee cut in % (rounded to two decimal places) for which the revenue does not shrink. Check your rounded value. [3 pts.]

Wrap-Up & 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.

  1. Write the exponential growth model \(N(t) = N_0 \cdot b^t\) (find \(b\))

  2. How many users will they have after 5 years?

  3. Is this discrete or continuous growth? What would the continuous model be?

  4. 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!