Problem 103 - A Graphical Approach to Precalculus with Limits

Question

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=7-5 \log x$$

Step-by-Step Solution

Answer

The solution to \( f(x) = 0 \) is \( x = 10^{\frac{7}{5}} \). For \( f(x)<0 \), \( x > 10^{\frac{7}{5}} \). For \( f(x) \geq 0 \), \( x \leq 10^{\frac{7}{5}} \).

Step 1: Solve the equation analytically

To solve the equation analytically, we begin with the given equation \( f(x) = 0 \) where \( f(x) = 7 - 5 \log x \). Set the function equal to zero: \[ 7 - 5 \log x = 0 \].Rearrange the equation to isolate the logarithmic term: \[ 5 \log x = 7 \].Now, divide both sides by 5 to solve for \( \log x \):\[ \log x = \frac{7}{5} \].To find \( x \), rewrite the logarithmic equation in its exponential form:\[ x = 10^{\frac{7}{5}} \].

Step 2: Graph the function

Consider the function \( y = f(x) = 7 - 5 \log x \). The graph of this function is a downward-shifting logarithmic curve, starting at high positive values and approaching negative infinity as \( x \) increases. The curve will intersect the x-axis at the point found in Step 1: \( x = 10^{\frac{7}{5}} \).

Step 3: Solve the inequality \(f(x)<0\) using the graph

To determine where \( f(x) < 0 \), inspect the graph of the function. \( f(x) \) is negative for values of \( x \) where the graph is below the x-axis. According to our solution, the graph is below the x-axis when \( x > 10^{\frac{7}{5}} \).

Step 4: Solve the inequality \(f(x) \geq 0\) using the graph

Now, identify where \( f(x) \geq 0 \). This occurs when the graph of the function is on or above the x-axis. According to our graph, this happens when \( x \leq 10^{\frac{7}{5}} \).

Key Concepts

Logarithmic Functions

Logarithmic functions are essential in many areas of mathematics and science, due to their characteristic growth behavior. At its core, a logarithmic function involves the concept of a logarithm, which is the inverse operation of exponentiation. When you see a function like \( f(x) = 7 - 5 \log x \), it tells us that we are dealing with a transformation of the basic logarithmic function \( \log x \).
One of the most important aspects of solving equations involving logarithmic functions is the ability to manipulate the logarithm. In this exercise, we took the equation \( 7 - 5 \log x = 0 \) and isolated the logarithm to get \( \log x = \frac{7}{5} \). This rearrangement is essential, since it allows us to rewrite this in exponential form: \( x = 10^{\frac{7}{5}} \), which is the solution to our equation.
Understanding how to switch between logarithmic and exponential forms is crucial, as many problems require this flexibility. It's also key to understand the properties of logarithms, like how they transform shifts in the curve, allowing us to interpret the function's behavior more intuitively.

Inequalities

Inequalities involve finding the range of values that satisfy a particular condition. In our problem, we are tasked with finding where the function \( f(x) = 7 - 5 \log x \) is either less than 0 or greater than or equal to 0.
To solve inequalities, a beneficial strategy is to employ a graphical analysis, which we will delve into later. Inequalities often require us to appreciate not just the single solution, but a set or range of solutions.
For example, we found that \( x > 10^{\frac{7}{5}} \) ensures \( f(x) < 0 \), indicating the portion of the graph where it dips below the x-axis. For the complementary inequality \( f(x) \geq 0 \), the solution \( x \leq 10^{\frac{7}{5}} \) tells us where the graph touches or remains at or above the x-axis. This approach of identifying critical points and corresponding test intervals is key to solving inequalities analytically.

Graphical Analysis

Graphical analysis is an intuitive way to understand functions and their solutions. By graphing the function \( f(x) = 7 - 5 \log x \), we get a visual representation of how the function behaves across different values of \( x \).
The function's graph starts high, representing positive values of \( y \), and decreases, approaching negativity to the right. The key takeaway from our graphical inspection is the x-intercept, which occurs at \( x = 10^{\frac{7}{5}} \). This intersect point is pivotal, as it marks the transition from non-negative values (\( f(x) \geq 0 \)) to negative values (\( f(x) < 0 \)).
Thus, by inspecting the graph, we quickly ascertain where \( f(x) < 0 \) and \( f(x) \geq 0 \) without solely relying on algebraic manipulation. Graphical analysis also helps verify our analytical solution, ensuring both approaches result in consistent interpretations.