Problem 99 - Precalculus : Building Concepts and Connections

Question

The graph of \(f(x)=a \log x\) passes through the point \((10,3) .\) Find \(a\) and thus the complete expression for \(f\) Check your answer by graphing \(f\).

Step-by-Step Solution

Answer

The constant \(a\) is approximately 1.43 and, thus, the complete expression for \(f\) is \(f(x) = 1.43 \log x\). The graph of \(f\) should pass through \((10,3)\) and confirm this result.

Step 1: Insert the Point Coordinates into the Equation

The given point \((10,3)\) can be written as \((x, f(x))\). The coordinates can then be inserted into the formula \(f(x) = a \log x\), resulting in the equation \(3 = a \log 10\).

Step 2: Solve the Equation for \(a\)

Isolate \(a\) to find its value by dividing both sides of the equation by \(\log 10\). This results in \(a = 3/\log 10\). When calculating through, you'll find that \(a \approx 1.43\).

Step 3: Write the Complete Expression for \(f\)

With \(a\) known, one can write down the full expression for \(f\), which will be \(f(x) = 1.43 \log x\).

Step 4: Graph the Function

Draw the graph of \(f\). The function should pass through the point \((10,3)\) and have a slope approximately equal to 1.43 at that point.

Key Concepts

Graphing Functions

Graphing functions can be a helpful way to see how mathematical expressions behave visually. In this case, we're mapping a logarithmic function, which has the general form of \(f(x) = a \log x\). The graph of this function has distinct characteristics:

• It passes through the point \((1,0)\) when \(aeq0\) because \(\log 1 = 0\).
• It's always increasing if \(a > 0\) since the log function increases as \(x\) increases.

To graph \(f(x) = 1.43 \log x\), you start by plotting the point \((10,3)\) since it lies on the graph based on the problem statement. Then, sketch the curve so that it extends gradually upward. Remember, the curve will approach zero as \(x\) approaches zero, but it never actually touches the y-axis because \(\log 0\) is undefined.

Finding Parameters

Finding parameters involves solving for unknowns in a function using given information. In this exercise, the parameter to be found is \(a\) in the expression \(f(x) = a \log x\). We know the function passes through the point \((10,3)\), meaning when \(x = 10\), \(f(x) = 3\).
To find \(a\), substitute these values into the equation:

• \(3 = a \log 10\)
• Solve for \(a\) to get \(a = \frac{3}{\log 10}\)

Remember that the value of \(\log 10\) is 1 when using base 10 logarithms. Thus, \(a\) simplifies to 3, making the expression for \(f\):

• \(f(x) = 3 \log x\)

Mathematical Proof

Mathematical proof is about validating the assumptions and ensuring the calculated outcomes are consistent with the known properties. In this problem, we've determined that \(a = \frac{3}{\log 10}\). To ensure this is indeed correct, understand that the logarithm of 10 in base 10 equals 1.
Thus, \(a = \frac{3}{1} = 3\). By checking back to our function \(f(x) = 3 \log x\), when we input \(x = 10\), the output is 3, which aligns with the point \((10,3)\) given in the problem.

• Substituting to verify: \(f(10) = 3 \log 10 = 3\)

This outcome substantiates the derived parameter and function as accurate and valid.

Function Evaluation

Function evaluation refers to determining the output of a function for a particular input. Based on the provided exercise, evaluate \(f(x)\) at specific values to verify accuracy. Now that we've established \(f(x) = 3 \log x\), let's evaluate some examples:

• When \(x = 1\), \(f(1) = 3 \log 1 = 3 \times 0 = 0\)
• When \(x = 10\), \(f(10) = 3 \log 10 = 3\)
• When \(x = 100\), \(f(100) = 3 \log 100 = 3 \times 2 = 6\)

Evaluating these values shows how the function behaves at different inputs and confirms that our calculations align with the characteristics of logarithmic functions.