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

What do you know about exponential and logarithmic functions? An exponential equation is a calculation in which the variable appears in an exponent. A logarithmic equation is a...

Questions: 10 | Attempts: 2624 | Last updated: Mar 22, 2023

Sample Question

Y = 2x is a(n)_____.

Exponential function

Logarithmic function

Power function

Polynomial function

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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

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Derivatives of Exponential and Logarithmic Functions Quiz Questions

Take up this quiz to review your knowledge of derivatives of exponential and logarithmic functions by choosing suitable answers to the questions asked here. The logarithmic...

Questions: 10 | Attempts: 1532 | Last updated: Apr 14, 2023

Sample Question

Differentiate y= 3e-4x

Dy/dx = -3e-3x

Dy/dx= -12e4x

Dy/dx= -12e-4x

Dy/dx = 34e4x

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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.

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Applications Of Exponential Functions Quiz

Do you know all about exponential functions? Take the quiz if you know where and how it can be used. This exponential functions quiz will help you understand your level of...

Questions: 15 | Attempts: 325 | Last updated: Aug 16, 2023

Sample Question

James decides to invest $20 in stock at Target. The stock raises 1.8% each year. Write an equation to model the situation then find how much money this stock will be worth in 6 years.

Y = 20 (0.018)x $1.11

Y = 20 (0.982)x $17.93

Y = 1.8 (20)x $115,200,000

Y = 20 (1.018)x $22.26

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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).