Problem 23 - Precalculus with Calculus Previews

Question

In Problems 23 and 24, find a logarithmic function \(f(x)=\log _{b} x\) such that the graph of \(f\) passes through the given point. $$ (49,2) $$

Step-by-Step Solution

Answer

The logarithmic function is \( f(x) = \log_7 x \).

Step 1: Identify Given Point

In this problem, we need to find the base, \( b \), of the logarithmic function \( f(x) = \log_b x \) such that it passes through the point (49, 2). The given point implies that when \( x = 49 \), \( f(x) = 2 \).

Step 2: Set Up the Logarithmic Equation

From the information provided, we know that \( f(49) = 2 \). This translates to the equation \( \log_b 49 = 2 \). We need to solve this equation to find the value of \( b \).

Step 3: Convert Logarithmic to Exponential Form

Using the property that \( \log_b A = C \) is equivalent to \( A = b^C \), convert \( \log_b 49 = 2 \) to exponential form: \( 49 = b^2 \).

Step 4: Solve for the Base \( b \)

Now solve the equation \( 49 = b^2 \) for \( b \). Taking the square root of both sides, we find that \( b = \sqrt{49} = 7 \). Since the base of a logarithm must be positive, \( b = 7 \) is our solution.

Key Concepts

Graphing Logarithmic Functions

Understanding how to graph logarithmic functions is essential for visualizing these mathematical expressions. A logarithmic function, typically of the form \(f(x) = \log_b x\), has a graph that is the inverse of its corresponding exponential function \(y = b^x\). The nature of this graph depends on the base \(b\):

• If \(b > 1\), the function is increasing, and its graph rises to the right, passing through the point \((1, 0)\).
• If \(0 < b < 1\), the function is decreasing, and its graph falls to the right, also passing through the point \((1, 0)\).

A key feature of logarithmic graphs is their vertical asymptote, which occurs at \(x = 0\). As \(x\) approaches 0 from the right, \(f(x)\) approaches negative infinity, but it never actually reaches \(x = 0\). This property highlights the importance of restricting the domain to positive \(x\) values. Additionally, logarithmic graphs are always continuous and smooth, devoid of any breaks or sharp turns, providing a clear visualization of the function's behavior.

Exponential and Logarithmic Equations

Solving exponential and logarithmic equations involves transforming one equation type into the other. This is because these two types of equations are inverse operations. In our example, we transformed the logarithmic equation \(\log_b 49 = 2\) into its equivalent exponential form \(49 = b^2\). This conversion process relies on the fundamental logarithmic identity:

• \(\log_b A = C\) is equivalent to \(A = b^C\)

By converting between these forms, you can utilize the familiar operations of powers and roots to solve for unknown variables like the base \(b\). Such conversions simplify the solution process by expressing exponential relationships in a more solvable format.
When solving exponential equations, remember to check if the results satisfy the original logarithmic equation, especially considering possible restrictions on the base \(b\) (it must be positive and not equal to 1). By understanding how to manipulate and interconvert these equations, solving complex problems becomes a straightforward process.

Properties of Logarithms

The properties of logarithms are tools that simplify complex logarithmic expressions and are crucial for solving equations efficiently. Let's explore some key properties:

• Product Property: \(\log_b (XY) = \log_b X + \log_b Y\)
• Quotient Property: \(\log_b \left(\frac{X}{Y}\right) = \log_b X - \log_b Y\)
• Power Property: \(\log_b (X^C) = C\log_b X\)

These properties allow us to transform logarithmic expressions into simpler forms, making it easier to solve equations or analyze function behavior. Knowing these properties enables the tackling of more advanced problems, as they provide the means to break down and reassemble exponential and logarithmic terms.
In solving our example, while these specific properties weren't explicitly used, understanding them is valuable, particularly if more complex points or equations were involved. These properties form the backbone of understanding and working with logarithmic functions, making them indispensable in both classroom learning and application in various fields.