Problem 100 - Precalculus : Building Concepts and Connections

Question

The graph of \(f(x)=A \ln x+B\) passes through the points (1,2) and \((e, 4)\) (a) Find \(A\) and \(B\) using the given points. (b) Check your answer by graphing \(f\)

Step-by-Step Solution

Answer

The values for A and B in the function \(f(x) = A \ln x + B\) are both 2 and the graph of the function validates these values.

Step 1: Substitute the first pair of points into the function

By substituting x = 1, and y = 2 into the function, an equation to find B is gotten. The function then is \(2 = A \ln 1 + B\). Because the \(\ln 1 = 0\), this simplifies to \(B = 2\).

Step 2: Substitute the second pair of points into the derived function

With B's value known, the value of A can now be found by substituting the second x-y pair, and B = 2 into the equation: \(4 = A \ln e + 2\). Using the fact that \(\ln e = 1\), the equation simplifies to \(A = 4 - 2 = 2\).

Step 3: Graph the function

Having determined that A = 2 and B = 2, the defined function is: \(f(x) = 2 \ln x + 2\). When this is graphed, it is seen that the function does cross at points (1,2) and (e,4), confirming the solution.

Key Concepts

Logarithmic Functions

Logarithmic functions are indispensable in mathematics, especially in contexts involving growth and scaling. A logarithmic function takes the form of \(f(x) = A \ln x + B\), where \(A\) and \(B\) are constants. The function \(\ln x\) represents the natural logarithm of \(x\), which is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. This function maps input numbers to their logarithmic scale, meaning it compresses large numbers and stretches small ones.
Logarithmic functions exhibit certain key properties. For instance, they are undefined for non-positive values of \(x\), which means the domain is \(x > 0\). The function is increasing, meaning as \(x\) increases, \(f(x)\) also increases. A logarithmic graph grows slower than linear or exponential graphs, making it particularly useful for modeling processes that reach saturation. For example, in our exercise, the expression \(2 \ln x + 2\) denotes a function that scales and shifts the standard logarithmic curve through transformation.

Graphing Techniques

Graphing techniques for logarithmic functions require a good understanding of their characteristics and transformations. To graph \(f(x) = A \ln x + B\), follow these steps:

• Identify transformations: The value \(A\) affects the vertical stretch or compression of the graph. If \(A\) is positive, the graph maintains its original orientation, multiplying its amplitude. The value \(B\) translates the entire graph vertically.
• Determine critical points: For example, with our function \(f(x)=2 \ln x + 2\), the point where \(x=1\) satisfies \(\ln 1 = 0\), allowing easy plotting. Similarly, \(x = e\) renders \(\ln e = 1\).
• Graph behavior: Note that for very small \(x\)-values, close to zero, the output of \(f(x)\) will approach negative infinity. However, as \(x\) becomes larger, the function will grow slowly and eventually outpace any linear growth due to the logarithmic nature.

The graph of \(f(x) = 2 \ln x + 2\) will cut through the given points, representing how any change in \(x\) modifies \(f(x)\) according to these transformations.

Problem Solving

Approaching problems involving logarithmic functions often follows a strategic path. Here's how to tackle these types of problems efficiently:

• Substitute known values: Begin by substituting given x-y pairs into the function, isolating one variable if possible. For instance, from the exercise, substituting \(x=1\) and \(y=2\) simplifies finding \(B\) since \(\ln 1 = 0\).
• Use known identities: Remember properties like \(\ln e = 1\) to simplify calculations, as in the second step of our example, finding \(A\).
• Reconfirm with graphing: After calculating values, graphing the function helps verify the solution. If the function's graph passes through the given points, your solution is likely correct.

The problem-solving process is clarified by this repetitive substitution and solving technique, where each calculation step helps build toward the complete function definition, as illustrated with \(f(x)=2 \ln x + 2\). The final graph serves as a confirmation tool, visually supporting your analytical results.