📈 Algebra: Exponential Functions

Growth, decay, and the power of e

🎯 Learning Objectives

  • Understand exponential functions and their properties
  • Graph exponential growth and decay functions
  • Solve exponential equations
  • Apply exponential functions to real-world problems
  • Understand logarithms as inverse functions
  • Use properties of exponents and logarithms

1. Understanding Exponential Functions

📈 Exponential Functions

An exponential function has the form:

f(x) = a \cdot b^x

Where:

  • a = initial value (y-intercept)
  • b = base (growth/decay factor)
  • x = exponent (time/input)

📈 Exponential Growth

b > 1 (increases over time)

Example: f(x) = 2ˣ

📉 Exponential Decay

0 < b < 1 (decreases over time)

Example: f(x) = (½)ˣ

🌟 Exponential Function Explorer

Current function: f(x) = 1 × 2ˣ
Y-intercept: 1
Behavior: Growth
Domain: All real numbers
Range: y > 0

Function Values:

2. Properties of Exponents

⚡ Exponent Rules

These rules help simplify exponential expressions:

Product Rule

a^m \cdot a^n = a^{m+n}

Same base: add exponents

Power Rule

(a^m)^n = a^{mn}

Power to a power: multiply exponents

Quotient Rule

\frac{a^m}{a^n} = a^{m-n}

Subtract exponents

Zero Rule

a^0 = 1

(a ≠ 0)

Negative Rule

a^{-n} = \frac{1}{a^n}

Reciprocal

Fraction Rule

a^{m/n} = \sqrt[n]{a^m}

Rational exponents = roots

⚡ Exponent Rules Practice

1. x⁵ × x³ =

2. (2³)⁴ =

3. x⁷ ÷ x⁴ =

4. 5⁰ =

3. Solving Exponential Equations

🎯 Solving Strategy

To solve exponential equations:

  1. Isolate the exponential expression
  2. Take logarithms of both sides (or use same base)
  3. Solve for the variable
  4. Check your solution

Example 1: Solve 2ˣ = 8

Step 1: Write both sides with same base: 2ˣ = 2³

Step 2: Set exponents equal: x = 3

Check: 2³ = 8 ✓

Example 2: Solve 3ˣ = 12

Step 1: Take natural log: ln(3ˣ) = ln(12)

Step 2: Use log property: x × ln(3) = ln(12)

Step 3: Solve: x = ln(12) ÷ ln(3) ≈ 2.262

Check: 3^2.262 ≈ 12 ✓

🧮 Exponential Equation Solver

Solve: ^ =

📝 Practice Solving

1. Solve: 4ˣ = 16

x =

2. Solve: 3ˣ = 27

x =

3. Solve: 5ˣ = 25

x =

4. Logarithms as Inverse Functions

🔄 Logarithms

Logarithms are the inverse of exponential functions.

Definition:

If b^y = x, then y = \log_b x

"y is the logarithm base b of x"

Examples:

  • \log_2 8 = 3 because 2³ = 8
  • \log_{10} 100 = 2 because 10² = 100
  • \ln e = 1 because e¹ = e
📊 Logarithm Calculator
=
📝 Logarithm Practice

1. log₂ 16 =

2. log₁₀ 1000 =

3. ln e² =

5. Real-World Exponential Applications

🌍 Exponential Growth & Decay

Exponential functions model many real-world phenomena:

💰 Compound Interest

A = P(1 + r/n)^(nt)

Where A = final amount, P = principal, r = rate, n = compounding frequency, t = time

🦠 Population Growth

P = P₀ e^(rt)

Exponential growth model

☢️ Radioactive Decay

N = N₀ e^(-λt)

Half-life calculations

🌡️ Newton's Law of Cooling

T = T₊ + (T₀ - T₊)e^(-kt)

Temperature change over time

💰 Compound Interest Calculator
📄 algebra-exponential.html | 2025-12-26