Constructing A Logarithmic Function F(x) With Specific Properties

In this article, we'll dive deep into the fascinating world of logarithmic functions and explore how to construct one that satisfies specific criteria. We're going to build a logarithmic function that exhibits a proportional change of 6.5 over equal-length output-value intervals, has an additive change of 2.2 between output values, and gracefully passes through the coordinate (3.6, 0). Buckle up, math enthusiasts, because we're about to embark on a mathematical journey!

Understanding Logarithmic Functions

Before we dive into the nitty-gritty details of constructing our function, let's take a moment to appreciate the beauty and versatility of logarithmic functions. These functions are the inverse of exponential functions, and they play a crucial role in various fields, including mathematics, physics, engineering, and even finance. Guys, think of them as the unsung heroes of the mathematical world!

A logarithmic function generally takes the form:

$$f(x) = a \log_b(x - c) + d$$

Where:

• f(x) represents the output value of the function for a given input x.
• a is a vertical stretch or compression factor.
• b is the base of the logarithm (which must be positive and not equal to 1).
• c is a horizontal shift.
• d is a vertical shift.

Understanding these parameters is key to manipulating and constructing logarithmic functions to fit our specific needs. The base b determines the rate at which the function grows or decays, while a, c, and d allow us to stretch, shift, and position the function in the coordinate plane. The horizontal shift c is particularly important as it affects the domain of the logarithmic function, ensuring that the argument of the logarithm (x - c) remains positive.

In our case, we need to determine the values of these parameters (a, b, c, and d) to satisfy the given conditions. We'll use the information about the proportional change, additive change, and the point (3.6, 0) to carefully piece together the puzzle and construct our desired logarithmic function. It's like being a mathematical detective, piecing together clues to solve a fascinating case!

Deconstructing the Given Conditions

To successfully construct our logarithmic function, we need to carefully analyze the conditions provided. These conditions are the breadcrumbs that will lead us to the solution. Let's break them down one by one:

1. Proportional change of 6.5 over equal-length output-value intervals: This tells us about the base of the logarithm. In simpler terms, for every equal increase in the output values (y-values), the input values (x-values) are multiplied by 6.5. This is a crucial piece of information that directly relates to the base b of our logarithmic function. It's like finding the key piece in a jigsaw puzzle!

2. Additive change of 2.2 between output values: This condition might seem a bit cryptic at first, but it gives us a crucial insight into the vertical stretching or compression of the function. An additive change in output values relates to the vertical scaling factor a. However, to fully utilize this condition, we need to carefully consider its relationship with the proportional change in input values. It's like deciphering a secret code within the problem!

3. Passes through the coordinate (3.6, 0): This is a specific point that lies on the graph of our logarithmic function. This point provides us with a concrete pair of (x, y) values that must satisfy the equation of our function. This condition acts as an anchor, grounding our function in the coordinate plane and allowing us to solve for one of the remaining unknown parameters. Think of it as a landmark guiding us on our mathematical journey!

By carefully deconstructing these conditions, we can begin to see how they relate to the parameters of our logarithmic function. It's like having a blueprint that outlines the steps we need to take to build our desired function. We'll use these conditions as our guide, meticulously working through each step to construct the perfect logarithmic function.

Determining the Base of the Logarithm (b)

Let's tackle the first condition: the proportional change of 6.5 over equal-length output-value intervals. As we discussed earlier, this condition directly relates to the base b of the logarithm. Remember, in a logarithmic function, the base determines how quickly the function grows or decays. In our case, we have a proportional change, meaning that for every fixed increment in the output (y-value), the input (x-value) is multiplied by a constant factor.

This constant factor is precisely the base of our logarithm! So, in this case, the base b is 6.5. This is a significant breakthrough, as we've already determined one of the key parameters of our function. It's like solving the first major puzzle in our mathematical quest!

With the base determined, our function now looks like this:

$$f(x) = a \log_{6.5}(x - c) + d$$

We've taken the first step in shaping our function, and we're well on our way to constructing the logarithmic function that meets all the given conditions. The next step is to decipher the remaining parameters, using the other clues we've been given. We're like skilled artisans, meticulously crafting our mathematical masterpiece, one step at a time.

Unveiling the Vertical Stretch (a)

Now, let's focus on the second condition: the additive change of 2.2 between output values. This condition tells us about how the function stretches or compresses vertically. Remember, the parameter a in our logarithmic function controls this vertical scaling. The key is to relate this additive change in output values to the logarithmic behavior dictated by the base we just found (6.5).

Let's consider two points on our logarithmic function, $(x_1, y_1)$ and $(x_2, y_2)$, where the output values differ by 2.2, i.e., $y_2 = y_1 + 2.2$. Due to the proportional change, we know that the corresponding input values must satisfy $x_2 = 6.5x_1$. This is the crucial link between the additive change in output and the multiplicative change in input, governed by the base of the logarithm.

We can write the function values at these two points as:

$$y_1 = a \log_{6.5}(x_1 - c) + d$$

$$y_2 = a \log_{6.5}(x_2 - c) + d$$

Substituting $y_2 = y_1 + 2.2$ and $x_2 = 6.5x_1$, we get:

$$y_1 + 2.2 = a \log_{6.5}(6.5x_1 - c) + d$$

Now, subtracting the first equation from this new equation, we get:

$$2.2 = a [\log_{6.5}(6.5x_1 - c) - \log_{6.5}(x_1 - c)]$$

This equation connects the vertical stretch a with the logarithmic relationship dictated by the base 6.5. However, to isolate a, we need to make a clever assumption. Let's assume that $c$ is negligible compared to $x_1$, so we can approximate $6.5x_1 - c \approx 6.5x_1$ and $x_1 - c \approx x_1$. This simplifies our equation to:

$$2.2 \approx a [\log_{6.5}(6.5x_1) - \log_{6.5}(x_1)]$$

$$2.2 \approx a [\log_{6.5}(6.5) + \log_{6.5}(x_1) - \log_{6.5}(x_1)]$$

$$2.2 \approx a$$

Therefore, we find that the vertical stretch factor a is approximately 2.2. This is another major step forward in our construction. We're meticulously uncovering each parameter, like a detective solving a complex puzzle. Our function now looks like:

$$f(x) = 2.2 \log_{6.5}(x - c) + d$$

We're getting closer to our goal, with only the horizontal shift c and the vertical shift d left to determine. The final piece of the puzzle awaits us!

Finding the Horizontal and Vertical Shifts (c and d)

Finally, let's use the third condition: the function passes through the coordinate (3.6, 0). This point gives us a specific (x, y) pair that must satisfy our function's equation. This is a golden opportunity to solve for the remaining unknowns, c and d. It's like finding the final piece of a jigsaw puzzle, the one that completes the picture!

Substituting x = 3.6 and f(x) = 0 into our current function, we get:

$$0 = 2.2 \log_{6.5}(3.6 - c) + d$$

We now have one equation with two unknowns. To solve for both c and d, we need another equation. This is where we make a crucial observation. Since logarithmic functions are defined for positive arguments, the term inside the logarithm (3.6 - c) must be greater than zero. This gives us a constraint on the possible values of c:

$$3.6 - c > 0$$

$$c < 3.6$$

This constraint is important, as it limits the range of possible horizontal shifts. However, it doesn't directly help us solve for c and d. To proceed, we'll make a strategic choice. Let's set the horizontal shift c such that the argument of the logarithm becomes 1. This simplifies the equation considerably, as the logarithm of 1 is always zero:

$$3.6 - c = 1$$

$$c = 2.6$$

This is a clever move! By setting c to 2.6, we've made the logarithmic term in our equation vanish, leaving us with a simple equation to solve for d:

$$0 = 2.2 \log_{6.5}(3.6 - 2.6) + d$$

$$0 = 2.2 \log_{6.5}(1) + d$$

$$0 = 2.2(0) + d$$

$$d = 0$$

And there we have it! We've found that the vertical shift d is 0. This completes our parameter search. We've meticulously uncovered each piece of the puzzle, and we're now ready to assemble the final logarithmic function.

The Grand Finale: Constructing the Logarithmic Function

After our mathematical journey, we've finally arrived at our destination: the complete logarithmic function that satisfies all the given conditions. Let's recap what we've found:

• Base (b): 6.5
• Vertical stretch (a): 2.2
• Horizontal shift (c): 2.6
• Vertical shift (d): 0

Plugging these values into our general logarithmic function form, we get:

$$f(x) = 2.2 \log_{6.5}(x - 2.6) + 0$$

Simplifying, our final logarithmic function is:

$$f(x) = 2.2 \log_{6.5}(x - 2.6)$$

This is the function we've been striving to construct! It exhibits a proportional change of 6.5 over equal-length output-value intervals, has an additive change related to the vertical stretch of 2.2, and passes gracefully through the coordinate (3.6, 0). We've successfully navigated the mathematical terrain and built a function that meets all the specified criteria.

Constructing this logarithmic function has been a rewarding experience. We've seen how the different parameters interact to shape the function's behavior, and we've used the given conditions as clues to uncover their values. This exercise demonstrates the power and versatility of logarithmic functions and the beauty of mathematical problem-solving. Remember guys, math is not just about formulas and equations; it's about logical thinking, problem-solving, and the joy of discovery!