Exponential Functions

📐 Exponential Growth

Part 1 of 7 — Exponential Growth

$y = a \cdot b^x quad (b > 1)$

• $a$ = initial value
• $b$ = growth factor
• Growth rate $r$: $b = 1 + r$

Exponential growth: each step multiplies by $b$.

Worked Example

Population: 500, grows 10% per year. After 3 years?

$y = 500(1.10)^3 = 500(1.331) = 665.5$ ✅

Concept Check 🎯

Exponential Growth 🧮

1. $100(1.2)^2 = ?$

2. $1000(1.5)^1 = ?$

3. $y = 2^x$. When $x = 3$, $y = ?$

Concept Check 🔍

Practice

Challenge Question 📋

📊 Exponential Decay

Part 2 of 7 — Exponential Decay

$y = a \cdot b^x quad (0 < b < 1)$

Decay factor: $b = 1 - r$ (where $r$ is the decay rate)

Half-life: time for the quantity to halve.

Worked Example

Car: $20,000, depreciates 15%/year. Value after 2 years?

$y = 20000(0.85)^2 = 20000(0.7225) = 14450$ → $14,450 ✅

Concept Check 🎯

Exponential Decay 🧮

1. $1000(0.9)^2 = ?$

2. $500(0.5)^1 = ?$

3. $800(0.75)^2 = ?$

Concept Check 🔍

Practice

Challenge Question 📋

🔢 Compound Interest

Part 3 of 7 — Compound Interest

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

• $P$ = principal, $r$ = annual rate, $n$ = compounding frequency, $t$ = years

Continuous compounding: $A = Pe^{rt}$

Worked Example

$1000 at 6% compounded annually for 2 years.

$A = 1000(1.06)^2 = 1000(1.1236) = 1123.6$ → $1,123.60 ✅

Concept Check 🎯

Compound Interest 🧮

1. $500 at 10%, 1 year, annual. A = ?

2. $1000 at 5%, 2 years, annual. A = ?

3. $1000 at 6%, 1 year, annual. A = ?

Concept Check 🔍

Practice

Challenge Question 📋

📈 Logarithms Introduction

Part 4 of 7 — Logarithms Introduction

A logarithm is the inverse of an exponential.

$log_b(x) = y iff b^y = x$

$\log_2(8) = 3$ because $2^3 = 8$

Common log: $\log(x) = log_{10}(x)$ Natural log: $ln(x) = log_e(x)$

Worked Example

$log_3(81) = ?$

$3^? = 81$. Since $3^4 = 81$, $log_3(81) = 4$ ✅

Concept Check 🎯

Evaluate Logs 🧮

1. $log_2(16) = ?$

2. $log_5(25) = ?$

3. $log_{10}(1000) = ?$

Concept Check 🔍

Practice

Challenge Question 📋

🧮 Log Properties

Part 5 of 7 — Log Properties

Key Properties

1. Product: $log_b(MN) = log_b(M) + log_b(N)$
2. Quotient: $log_b\left(\frac{M}{N}\right) = log_b(M) - log_b(N)$
3. Power: $log_b(M^p) = p \cdot log_b(M)$

Worked Example

$log_2(4 \cdot 8) = log_2(4) + log_2(8) = 2 + 3 = 5$ ✅

Check: $4 \times 8 = 32 = 2^5$ ✅

Concept Check 🎯

Log Properties 🧮

1. $log_2(4) + log_2(8) = ?$

2. $log_{10}(100) - log_{10}(10) = ?$

3. $log_{10}(10^3) = ?$

Concept Check 🔍

Practice

Challenge Question 📋

🛠️ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Apply exponential and logarithmic skills:

• Doubling time problems
• Real-world decay (depreciation, radioactive)
• Solving exponential equations with logs

Worked Example

$1000 at 8% annual. When does it double? (Rule of 72)

$72 \div 8 = 9$ years (approximately) ✅

Concept Check 🎯

Applications 🧮

1. Rule of 72: doubling at 6%. Years?

2. $1000(1.05)^2 = ?$

3. $log_2(64) = ?$

Concept Check 🔍

Practice

Challenge Question 📋

🏆 Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

• Growth: $y = a(1+r)^t$, Decay: $y = a(1-r)^t$
• Compound interest: $A = P(1+r/n)^{nt}$
• $log_b(x) = y iff b^y = x$
• Product/Quotient/Power rules for logs

Worked Example

$log_3(27) + log_3(9) = 3 + 2 = 5 = log_3(243)$ ✅

Concept Check 🎯

Review 🧮

1. $500(1.1)^2 = ?$

2. $1000(0.5)^3 = ?$

3. $log_3(27) = ?$

Concept Check 🔍

Practice

Challenge Question 📋