EXPERIMENTING with Exponential Function

Word problems involving exponential and logarithmic functions often require finding the values of unknown parameters. At HappyMath, we believe that mastering these techniques is essential for success in advanced mathematics. With the right approach, even complex problems become manageable.

In this guide, we'll explore systematic methods for solving word problems that involve finding parameters in exponential and logarithmic functions using given points and other information.

EXPERIMENTING with Exponential Function

Finding Parameters in Exponential Functions of the Form y = ae^x + b

One common form of exponential function is y = ae^x + b, where:

• 'a' is the vertical scaling factor
• 'b' is the vertical shift (determining the horizontal asymptote)

Let's explore a systematic approach to find these parameters when given specific points on the curve.

Example 1: Using Simultaneous Equations with Exponential Functions

Given an exponential function y = ae^x + b passing through points (0, 5) and (4, 11), find the values of a and b.

Step 1: Substitute the first point (0, 5) into the function.

• 5 = ae^0 + b
• 5 = a + b (since e^0 = 1)

Step 2: Substitute the second point (4, 11) into the function.

• 11 = ae^4 + b

Step 3: Set up a system of equations.

• a + b = 5 (Equation 1)
• ae^4 + b = 11 (Equation 2)

Step 4: Eliminate one variable by subtracting Equation 1 from Equation 2.

• ae^4 + b - (a + b) = 11 - 5
• ae^4 - a = 6
• a(e^4 - 1) = 6
• a = 6/(e^4 - 1)

Step 5: Substitute this value of a back into Equation 1 to find b.

• b = 5 - a = 5 - 6/(e^4 - 1)
• b = (5(e^4 - 1) - 6)/(e^4 - 1) = (5e^4 - 5 - 6)/(e^4 - 1) = (5e^4 - 11)/(e^4 - 1)

Therefore, a = 6/(e^4 - 1) and b = (5e^4 - 11)/(e^4 - 1).

This method allows us to find exact values for a and b, which can then be used to fully define the exponential function.

Finding Parameters in Logarithmic Functions of the Form y = a ln(x) + b

Logarithmic functions of the form y = a ln(x) + b have two parameters:

• 'a' affects the steepness of the curve
• 'b' represents the vertical shift

Example 2: Using the Zero Product Property in Logarithmic Functions

Given a logarithmic function y = a ln(x) + b passing through points (5, 0) and (10, 2), find the values of a and b.

Step 1: Substitute the first point (5, 0) into the function.

• 0 = a ln(5) + b

Step 2: Recognize that this equation can be satisfied in two ways:

• Either a = 0, or
• ln(5) + b/a = 0, which means b = -a ln(5)

Step 3: Substitute the second point (10, 2) into the function.

• 2 = a ln(10) + b

Step 4: Rule out the possibility that a = 0.

• If a = 0, then from the second point, we would have 2 = 0 + b, implying b = 2.
• But then from the first point, we would have 0 = 0 + 2, which is false.
• Therefore, a ≠ 0.

Step 5: Since a ≠ 0, we must have b = -a ln(5) from Step 2.

• Substitute this into the equation from Step 3:
• 2 = a ln(10) - a ln(5)
• 2 = a (ln(10) - ln(5))
• 2 = a ln(10/5)
• 2 = a ln(2)
• a = 2/ln(2)

Step 6: Find b by substituting back.

• b = -a ln(5) = -(2/ln(2)) ln(5) = -2 ln(5)/ln(2)

Therefore, a = 2/ln(2) and b = -2 ln(5)/ln(2).

This example demonstrates how to carefully navigate the zero product property when dealing with logarithmic equations.

Using Graphical Features to Identify Parameters

Sometimes, we can use features of a graph to quickly identify parameter values, reducing the algebraic workload.

Example 3: Using Asymptotes to Find Parameters

Given a graph of y = ae^x + b with a horizontal asymptote at y = 4 and passing through the point (0, 6), find the values of a and b.

Step 1: Identify the horizontal asymptote.

• For a function of the form y = ae^x + b, the horizontal asymptote as x → -∞ is y = b.
• Since the asymptote is at y = 4, we know that b = 4.

Step 2: Substitute the point (0, 6) and the value of b into the function.

• 6 = ae^0 + 4
• 6 = a + 4
• a = 2

Therefore, the function is y = 2e^x + 4.

This approach leverages our understanding of the graphical behavior of exponential functions to quickly identify parameters without solving complex systems of equations.

Solving Problems with Parameters in Both Exponential and Logarithmic Forms

Some advanced problems may require converting between exponential and logarithmic forms to solve for parameters efficiently.

Example 4: Converting Between Forms to Find Parameters

Given a function that can be written as either y = ae^x + b or y = c ln(x) + d, and passing through points (1, 5) and (e, 6), find all parameters.

Step 1: Set up equations for both forms using the given points.

• For the exponential form:
  
  
  • 5 = ae^1 + b = ae + b
  • 6 = ae^e + b
• For the logarithmic form:
  
  
  • 5 = c ln(1) + d = 0 + d, so d = 5
  • 6 = c ln(e) + d = c + 5, so c = 1

Step 2: Determine which form is valid by checking consistency.

• The logarithmic form gives us c = 1 and d = 5, so y = ln(x) + 5.
• Let's verify this with our points:
  • For (1, 5): 5 = ln(1) + 5 = 0 + 5 = 5 ✓
  • For (e, 6): 6 = ln(e) + 5 = 1 + 5 = 6 ✓

Therefore, the function is y = ln(x) + 5.

Key Strategies for Solving Word Problems with Exponential and Logarithmic Functions

When tackling word problems involving exponential and logarithmic functions, remember these key strategies:

1. Systematically substitute given points into the function equation to create a system of equations.
   
    
2. Look for graphical features like asymptotes that provide immediate information about parameters.
   
    
3. Consider special properties of exponential and logarithmic functions, such as e^0 = 1 and ln(1) = 0.
   
    
4. Check for degenerate cases – sometimes a parameter might be zero, leading to a simplified function.
   
    
5. Verify your solution by substituting it back into the original conditions.
   
    
6. Be precise with algebraic manipulations, especially when dealing with logarithm properties.
   
    

Building Your Problem-Solving Toolkit

At HappyMath, we emphasize that mastering these techniques provides valuable tools for modeling real-world phenomena such as population growth, radioactive decay, compound interest, and sound intensity.

With practice, you'll develop the confidence to approach even the most challenging word problems involving exponential and logarithmic functions. Remember that these problems often have multiple solution paths—developing the intuition to choose the most efficient approach is part of the mathematical journey.

By breaking down complex problems into manageable steps and applying systematic techniques, you'll find that what once seemed daunting becomes an opportunity to demonstrate your mathematical prowess and analytical thinking skills.