🎯⭐ INTERACTIVE LESSON

Exponential Functions

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Exponential Functions - Complete Interactive Lesson

Part 1: Exponential Growth

📐 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 🧮

$100(1.2)^2 = ?$
$1000(1.5)^1 = ?$

Concept Check 🔍

Practice

#	Initial	Rate	Years	Formula
1	100	20%	2	100(1.2)²
2	200	5%	3	200(1.05)³
3	1000	50%	1	1000(1.5)¹

Challenge Question 📋

Part 2: Exponential Decay

📊 Exponential Decay

Part 2 of 7 — Exponential Decay

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

Part 3: Compound Interest

🔢 Compound Interest

Part 3 of 7 — Compound Interest

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

Part 4: Logarithms Introduction

📈 Logarithms Introduction

Part 4 of 7 — Logarithms Introduction

A logarithm is the inverse of an exponential.

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

Part 5: Log Properties

🧮 Log Properties

Part 5 of 7 — Log Properties

Key Properties

Product: $log_b(MN) = log_b(M) + log_b(N)$

Part 6: Problem-Solving Workshop

🛠️ 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 🧮

Rule of 72: doubling at 6%. Years?
$1000(1.05)^2 = ?$

Part 7: Review & Applications

🏆 Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

Growth: $y = a(1+r)^t$ , Decay: $y = a(1-r)^t$

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

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 🧮

$1000(0.9)^2 = ?$
$500(0.5)^1 = ?$
$800(0.75)^2 = ?$

Concept Check 🔍

Practice

#	Initial	Rate	Years	Formula
1	1000	10%	2	1000(0.9)²
2	500	50%	1	500(0.5)¹
3	800	25%	2	800(0.75)²

Challenge Question 📋

$P$ = 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 🧮

$500 at 10%, 1 year, annual. A = ?
$1000 at 5%, 2 years, annual. A = ?
$1000 at 6%, 1 year, annual. A = ?

Concept Check 🔍

Practice

#	P	r	n	t	A
1	$1000	5%	1	2	$1102.50
2	$500	10%	1	1	$550
3	$2000	4%	1	3	$2249.73

Challenge Question 📋

$\log_2(8) = 3$ 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 🧮

$log_2(16) = ?$
$log_5(25) = ?$
$log_{10}(1000) = ?$

Concept Check 🔍

Practice

#	Log	Value
1	$log_2(16)$	4
2	$log_5(25)$	2
3	$log_{10}(1000)$	3

Challenge Question 📋

Quotient:

log_bleft(\frac{M}{N}\right) = log_b(M) - log_b(N)

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 🧮

$log_2(4) + log_2(8) = ?$
$log_{10}(100) - log_{10}(10) = ?$
$log_{10}(10^3) = ?$

Concept Check 🔍

Practice

#	Property	Example
1	Product	$log(2)+log(5) = log(10) = 1$
2	Quotient	$log(100)-log(10) = log(10) = 1$
3	Power	$log(10^3) = 3$

Challenge Question 📋

Concept Check 🔍

Practice

#	Problem	Answer
1	Doubling at 6%?	~12 years
2	$1000 ×(1.05)²	1102.5
3	log₂(64)	6

Challenge Question 📋

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 🧮

$500(1.1)^2 = ?$
$1000(0.5)^3 = ?$
$log_3(27) = ?$

Concept Check 🔍

Practice

#	Type	Problem
1	Growth	500(1.1)²
2	Decay	1000(0.5)³
3	Log	log₃(27)

Challenge Question 📋