Exponential and Logarithmic Functions Lesson - Rules & Examples

Lesson Overview

• What Are Exponential Functions and Logarithmic Functions?
• Relationship Between Exponential and Logarithmic Functions
• Graphical Explanation of Exponential and Logarithmic Functions
• Rules of Exponential Functions and Logarithmic Functions
• Exponential and Logarithmic Functions Examples
• Exponential and Logarithmic Functions Assessment

What Are Exponential Functions and Logarithmic Functions?

Exponential Functions

An exponential function is a mathematical function where the variable appears as an exponent. It has the general form:

f(x) = a * bˣ

• a is the initial value (constant),
• b is the base (b > 0, b ≠ 1),
• x is the exponent (variable).

Example:

If f(x) = 2ˣ:

• For x = 1, f(1) = 2¹ = 2.
• For x = 2, f(2) = 2² = 4.
• For x = -1, f(-1) = 2⁻¹ = 1/2 = 0.5.

The function grows rapidly as x increases.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It answers the question: "To what power must the base be raised to get the given number?"

The general form is:

f(x) = logₐ(x)

• a is the base (a > 0, a ≠ 1),
• x is the number (x > 0).

Example:

If log₂(8):

• 2 must be raised to the power of 3 to equal 8.
• Therefore, log₂(8) = 3 because 2³ = 8.

Take The Quiz :

Exponential and Logarithmic Functions! Math Trivia Quiz

Relationship Between Exponential and Logarithmic Functions

Graphical Explanation of Exponential and Logarithmic Functions

Graph of Exponential Function f(x) = 2ˣ:

• Passes through (0, 1).
• Rises quickly for positive x.
• Approaches 0 as x → -∞.

Fig: Graph of Exponential Function

Graph of Logarithmic Function g(x) = log₂(x):

• Passes through (1, 0).
• Increases slowly for large x.
• Undefined for x ≤ 0.

Fig: Graph of Logarithmic Function

Take The Quiz :

Derivatives of Exponential and Logarithmic Functions Quiz Questions

Rules of Exponential Functions and Logarithmic Functions

Rules of Exponential Functions

1. Product Rule
   bˣ * bʸ = bˣ⁺ʸ
   • Example: 2³ * 2² = 2³⁺² = 2⁵ = 32.
2. Quotient Rule
   bˣ ÷ bʸ = bˣ⁻ʸ
   • Example: 3⁵ ÷ 3² = 3⁵⁻² = 3³ = 27.
3. Power Rule
   (bˣ)ʸ = bˣʸ
   • Example: (2²)³ = 2²*³ = 2⁶ = 64.
4. Zero Exponent Rule
   b⁰ = 1 (where b ≠ 0)
   • Example: 5⁰ = 1.
5. Negative Exponent Rule
   b⁻ˣ = 1 / bˣ
   • Example: 2⁻³ = 1 / 2³ = 1/8.
6. Base 1 Rule
   1ˣ = 1
   • Example: 1² = 1.

Rules of Logarithmic Functions

1. Product Rule
   log_b(xy) = log_b(x) + log_b(y)
   • Example: log₂(8 * 4) = log₂(8) + log₂(4) = 3 + 2 = 5.
2. Quotient Rule
   log_b(x / y) = log_b(x) - log_b(y)
   • Example: log₃(27 / 3) = log₃(27) - log₃(3) = 3 - 1 = 2.
3. Power Rule
   log_b(xⁿ) = n * log_b(x)
   • Example: log₂(4³) = 3 * log₂(4) = 3 * 2 = 6.
4. Base Change Rule
   log_b(x) = log_k(x) / log_k(b)
   • Example: log₃(9) = log₁₀(9) / log₁₀(3) = 2.
5. Identity Rule
   log_b(b) = 1
   • Example: log₄(4) = 1.
6. Inverse Rule
   b^(log_b(x)) = x and log_b(bˣ) = x
   • Example: 2^(log₂(8)) = 8 and log₂(2³) = 3.
     
     

Exponential and Logarithmic Functions Examples

Example 1: Exponential Function
f(x) = 3²

• When x = 2, f(2) = 3² = 9.

Example 2: Logarithmic Function
f(x) = log₂(8)

• log₂(8) = 3 because 2³ = 8.

Example 3: Real-Life Application

• Exponential Growth: Population growth (P = P₀e^rt).
• Logarithms: Measuring sound (decibels) or pH levels.

Take The Quiz :

Applications Of Exponential Functions Quiz

Exponential and Logarithmic Functions Assessment

Practice Questions:

1. Find the value of 2³.
2. Solve for x: log₃(27) = x.
3. Sketch the graph of y = 2ˣ and y = log₂(x).
4. Simplify: log₄(64).
5. Calculate: 3⁵ / 3³.

Answers:

1. 8.
2. 3 (because 3³ = 27).
3. (Graph).
4. 3 (because 4³ = 64).
5. 9 (because 3⁵ / 3³ = 3² = 9).