Exponential And Logarithmic Functions Examples With Solutions

Exponential and Logarithmic Functions: Examples with Solutions

Exponential and logarithmic functions are fundamental concepts in mathematics with wide-ranging applications in various fields, including science, engineering, finance, and computer science. Understanding these functions, their properties, and how to solve problems involving them is crucial for success in many areas. This comprehensive guide will explore exponential and logarithmic functions, providing numerous examples with detailed solutions to solidify your understanding.

Understanding Exponential Functions

An exponential function is a function of the form f(x) = a^x, where 'a' is a positive constant called the base, and 'x' is the exponent. The key characteristic of an exponential function is that the independent variable (x) appears as an exponent.

Key Properties of Exponential Functions:

• Base: The base, 'a', must be a positive number and not equal to 1 (a > 0, a ≠ 1). If the base is 1, the function becomes a constant function, f(x) = 1.
• Growth/Decay: If a > 1, the function represents exponential growth. If 0 < a < 1, the function represents exponential decay.
• y-intercept: The y-intercept is always (0, 1) because a⁰ = 1 for any a ≠ 0.
• Asymptote: The x-axis (y = 0) acts as a horizontal asymptote for exponential functions. The graph approaches but never touches the x-axis.
• One-to-one: Exponential functions are one-to-one, meaning that each x-value corresponds to a unique y-value, and vice-versa. This allows for the existence of inverse functions.

Examples of Exponential Functions and their Solutions:

Example 1: Simple Growth

A population of bacteria doubles every hour. If the initial population is 1000, find the population after 5 hours.

Solution:

The formula for exponential growth is: P(t) = P₀ * a^t

Where:

• P(t) = population at time t
• P₀ = initial population (1000)
• a = growth factor (2, since it doubles)
• t = time (in hours)

P(5) = 1000 * 2⁵ = 1000 * 32 = 32000

Therefore, the population after 5 hours will be 32,000.

Example 2: Compound Interest

$1000 is invested at an annual interest rate of 5%, compounded annually. Find the balance after 10 years.

Solution:

The formula for compound interest is: A = P (1 + r/n)^(nt)

Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial deposit or loan amount)
• r = the annual interest rate (decimal)
• n = the number of times that interest is compounded per year
• t = the number of years the money is invested or borrowed for

In this case: P = 1000, r = 0.05, n = 1 (compounded annually), t = 10

A = 1000 (1 + 0.05/1)^(1*10) = 1000 (1.05)^10 ≈ 1628.89

The balance after 10 years will be approximately $1628.89.

Example 3: Exponential Decay

The half-life of a radioactive substance is 5 years. If you start with 100 grams, how much remains after 15 years?

Solution:

The formula for exponential decay is: A(t) = A₀ * (1/2)^(t/h)

Where:

• A(t) = amount remaining after time t
• A₀ = initial amount (100 grams)
• t = time elapsed (15 years)
• h = half-life (5 years)

A(15) = 100 * (1/2)^(15/5) = 100 * (1/2)³ = 100 * (1/8) = 12.5

After 15 years, 12.5 grams will remain.

Understanding Logarithmic Functions

A logarithmic function is the inverse of an exponential function. If y = aˣ, then the logarithmic function is written as x = logₐy. This is read as "x is the logarithm of y to the base a."

Key Properties of Logarithmic Functions:

• Base: The base 'a' must be a positive number and not equal to 1 (a > 0, a ≠ 1).
• Inverse Relationship: Logarithmic functions are the inverse of exponential functions. This means that logₐ(aˣ) = x and a^(logₐx) = x.
• Domain and Range: The domain of logₐx is (0, ∞), and the range is (-∞, ∞). Logarithms are only defined for positive arguments.
• Common Logarithm: The common logarithm (log x) has a base of 10 (log₁₀x).
• Natural Logarithm: The natural logarithm (ln x) has a base of e (Euler's number, approximately 2.71828).

Examples of Logarithmic Functions and their Solutions:

Example 1: Solving a Logarithmic Equation

Solve for x: log₂(x) = 3

Solution:

By definition of logarithm, this equation is equivalent to: 2³ = x

Therefore, x = 8.

Example 2: Solving a Logarithmic Equation with a Different Base

Solve for x: log₃(x + 2) = 2

Solution:

Rewrite the equation in exponential form: 3² = x + 2

9 = x + 2

x = 7

Example 3: Using Properties of Logarithms

Simplify the expression: log₄(16) + log₄(64)

Solution:

Using the property logₐ(mn) = logₐm + logₐn, we can simplify:

log₄(16) + log₄(64) = log₄(16 * 64) = log₄(1024)

Since 4⁵ = 1024, log₄(1024) = 5

Example 4: Change of Base Formula

Evaluate log₂(7) using a calculator (most calculators only have log₁₀ and ln).

Solution:

Use the change of base formula: logₐ(b) = logₓ(b) / logₓ(a), where x can be any base (usually 10 or e).

log₂(7) = log₁₀(7) / log₁₀(2) ≈ 2.807

Example 5: Real-world application - pH Calculation

The pH of a solution is given by the formula pH = -log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration. If the pH of a solution is 4, find the hydrogen ion concentration.

Solution:

4 = -log₁₀[H⁺]

-4 = log₁₀[H⁺]

10⁻⁴ = [H⁺]

Therefore, the hydrogen ion concentration is 10⁻⁴ moles per liter.

Applications of Exponential and Logarithmic Functions

The applications of exponential and logarithmic functions are vast and span numerous disciplines:

• Finance: Compound interest, loan calculations, and investment growth modeling.
• Biology: Population growth, radioactive decay, and drug absorption/elimination.
• Physics: Radioactive decay, cooling/heating processes, and wave phenomena.
• Chemistry: Reaction rates, pH calculations, and chemical kinetics.
• Computer Science: Algorithm analysis, data structures, and cryptography.
• Engineering: Signal processing, circuit analysis, and control systems.

This guide provides a strong foundation in understanding and applying exponential and logarithmic functions. By working through these examples and understanding the properties of these functions, you'll be well-equipped to tackle more complex problems in various fields. Remember to practice regularly to reinforce your knowledge and build confidence in solving problems involving these important mathematical concepts. Further exploration into more advanced topics like derivatives and integrals of these functions will deepen your understanding even further.