Logarithmic Function: Calculate 'a' And 'b'

In the realm of mathematics, understanding and manipulating functions is a cornerstone of problem-solving. Today, we're diving deep into the world of logarithmic functions, specifically one of the form $y=a+b \ln (x-2)$. Our mission, should we choose to accept it, is to determine the values of $a$ and $b$ when this function passes through two distinct points: $(3,-19)$ and $(495,105)$. We'll need to be precise, rounding our final answers to three decimal places if necessary. This isn't just about crunching numbers; it's about understanding how specific points constrain the behavior of a mathematical function. Let's break down this problem step-by-step, ensuring clarity and accuracy throughout our journey.

Understanding the Logarithmic Function Form

Before we get our hands dirty with calculations, let's take a moment to appreciate the structure of the logarithmic function we're working with: $y=a+b \ln (x-2)$. This equation represents a transformation of the basic natural logarithm function, $\ln(x)$. The parameter $b$ controls the vertical stretch or compression of the graph, and it also determines whether the graph is reflected across the x-axis (if $b$ is negative). The constant $a$ represents a vertical shift of the graph. The term $(x-2)$ inside the logarithm indicates a horizontal shift of the graph 2 units to the right. The natural logarithm, $\ln(x)$, is defined only for positive values of $x$. Therefore, for our function $y=a+b \ln (x-2)$, the expression $(x-2)$ must be positive, which means $x > 2$. This domain restriction is crucial to remember as we work with our given points. The points $(3,-19)$ and $(495,105)$ both satisfy this condition, as $3 > 2$ and $495 > 2$. The presence of two points is key because it gives us two independent equations, allowing us to solve for the two unknown constants, $a$ and $b$. This is a common strategy in mathematics: when you have as many equations as you have unknowns, you can often find a unique solution.

Setting Up the Equations

Our journey begins with the fundamental principle that if a function passes through a point, then the coordinates of that point must satisfy the function's equation. We are given two points, $(3,-19)$ and $(495,105)$, and our function is $y=a+b \ln (x-2)$. Let's substitute the coordinates of each point into the equation to create a system of two linear equations with two variables, $a$ and $b$.

For the first point, $(3,-19)$, we substitute $x=3$ and $y=-19$ into the equation:

$-19 = a + b \ln (3-2)$ $-19 = a + b \ln (1)$

Since the natural logarithm of 1 is 0 (because $e^0 = 1$), this simplifies to:

$-19 = a + b(0)$ $-19 = a$

This is a fantastic start! We've already determined the value of $a$ directly from the first point. This often happens when one of the points provides a particularly simple input for the logarithmic term, like in this case where $x-2=1$.

Now, let's use the second point, $(495,105)$. We substitute $x=495$ and $y=105$ into the equation:

$105 = a + b \ln (495-2)$ $105 = a + b \ln (493)$

We now have a system of two equations:

1. $a = -19$
2. $105 = a + b \ln (493)$

We can see that our task has been greatly simplified because we already know the value of $a$. We just need to substitute this value into the second equation to solve for $b$. This systematic approach ensures that we are building upon known facts and moving logically towards our solution.

Solving for 'b'

With the value of $a$ firmly established as $-19$, we can now substitute this into our second equation derived from the point $(495,105)$: $105 = a + b \ln (493)$.

Substituting $a = -19$ gives us:

$105 = -19 + b \ln (493)$

Our goal now is to isolate $b$. First, let's add 19 to both sides of the equation to move the constant term to the left side:

$105 + 19 = b \ln (493)$ $124 = b \ln (493)$

To find $b$, we need to divide both sides by $\ln (493)$. Before we do that, let's recall what $\ln (493)$ represents. It's the power to which the mathematical constant $e$ (approximately 2.71828) must be raised to equal 493. This value will be a positive number.

Now, we perform the division:

$b = \frac{124}{\ln (493)}$

To get a numerical value for $b$, we need to calculate $\ln (493)$ using a calculator. Ensure your calculator is set to natural logarithm (ln).

$\ln (493) \approx 6.19840313$

Now, we divide 124 by this value:

$b \approx \frac{124}{6.19840313}$ $b \approx 20.005124$

We are asked to round all values to three decimal places. Therefore, the value of $b$ rounded to three decimal places is:

$b \approx 20.005$

So, we have found both constants: $a = -19$ and $b \approx 20.005$. The function that passes through the given points is approximately $y = -19 + 20.005 \ln (x-2)$. This process of substitution and algebraic manipulation is fundamental in solving systems of equations that arise from function analysis. The precision required in the final answer highlights the importance of careful calculation and rounding.

Verification and Conclusion

We have determined that $a = -19$ and $b \approx 20.005$. To ensure our calculations are correct, let's plug these values back into the original function form, $y=a+b \ln (x-2)$, and check if it holds true for both given points. This verification step is crucial in mathematics to confirm that our derived solution accurately satisfies all the conditions of the problem.

Checking Point 1: (3, -19)

Using our found values: $a = -19$ and $b \approx 20.005$.

$y = -19 + 20.005 \ln (3-2)$ $y = -19 + 20.005 \ln (1)$ $y = -19 + 20.005 (0)$ $y = -19$

This matches the given y-coordinate of the first point, $-19$. So, the first point is verified.

Checking Point 2: (495, 105)

Using our found values: $a = -19$ and $b \approx 20.005$.

$y = -19 + 20.005 \ln (495-2)$ $y = -19 + 20.005 \ln (493)$

Now, we need to calculate $\ln (493)$. As we found earlier, $\ln (493) \approx 6.19840313$.

$y \approx -19 + 20.005 \times 6.19840313$ $y \approx -19 + 124.001027$ $y \approx 105.001027$

When we round this to three decimal places, we get $y \approx 105.001$. The given y-coordinate for the second point is $105$. The slight difference is due to the rounding of $b$ to three decimal places. If we used the more precise value of $b \approx 20.005124$, the result would be even closer to $105$. This level of accuracy confirms our calculated values for $a$ and $b$ are correct within the specified rounding.

In conclusion, by systematically substituting the given points into the logarithmic function $y=a+b \ln (x-2)$, we successfully determined the values of the constants $a$ and $b$. We found that $a = -19$ and $b \approx 20.005$. This problem beautifully illustrates how specific data points can define the parameters of a function, a fundamental concept in mathematical modeling and data analysis. Understanding these principles allows us to interpret and predict the behavior of systems described by logarithmic relationships.

For further exploration into logarithmic functions and their properties, you can visit websites like Khan Academy. They offer comprehensive resources and practice problems that can deepen your understanding of these essential mathematical concepts.