Problem 103 - A Graphical Approach to College Algebra

Question

For the given \(f(x)\), 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

\(x = 10^{\frac{7}{5}}\), \(f(x) < 0\) for \(x > 10^{\frac{7}{5}}\), \(f(x) \geq 0\) for \(0 < x \leq 10^{\frac{7}{5}}\).

Step 1: Solve the Equation Analytically

First, equate the given function to zero: \( 7 - 5 \log{x} = 0 \). Solve for \( x \) by isolating the logarithm: \( 5 \log{x} = 7 \). Divide both sides by 5: \( \log{x} = \frac{7}{5} \). To solve for \( x \), rewrite the equation in exponential form: \( x = 10^{\frac{7}{5}} \).

Step 2: Graph the Function

Sketch or use a graph of the function \( y = 7 - 5 \log{x} \). This function will have one vertical asymptote at \( x = 0 \). The function intersects the x-axis at the point \( x = 10^{\frac{7}{5}} \), which we found in Step 1.

Step 3: Determine the Inequality \(f(x) < 0\)

Inspect the graph where the function is below the x-axis. Since \( y = 7 - 5 \log{x} \) crosses the x-axis at \( x = 10^{\frac{7}{5}} \), the function is negative to the right of this point. Therefore, \( f(x) < 0 \) for \( x > 10^{\frac{7}{5}} \).

Step 4: Determine the Inequality \(f(x) \geq 0\)

From the graph, observe where the function is above or on the x-axis. Since \( f(x) = 0 \) at \( x = 10^{\frac{7}{5}} \) and \( f(x) \) is positive when \( x < 10^{\frac{7}{5}} \), we have \( f(x) \geq 0 \) for \( x \leq 10^{\frac{7}{5}} \) and \( x > 0 \).

Key Concepts

Graphical Analysis

Graphical analysis helps us to visualize equations and inequalities. In this exercise, we start by plotting the function \( y = 7 - 5 \log{x} \). Visualizing this function allows us to see important aspects like intercepts and regions where the function is above or below the x-axis. To do this effectively:

• Identify and mark the vertical asymptote at \( x = 0 \) because logarithmic functions tend toward negative infinity as \( x \to 0 \).
• Plot the intercept \( x = 10^{\frac{7}{5}} \), where the graph crosses the x-axis. This is critical because it determines where the function changes sign.

By analyzing these features, you can understand the behavior of the function: where it is positive (above the x-axis) and where it is negative (below the x-axis). Such insights make it possible to determine the solution to inequalities by simply inspecting the graph. Regularly using graphical analysis in conjunction with algebraic solutions enhances comprehension of how functions behave across different values of \( x \).

Inequalities

Inequalities involve finding regions where a function is greater than or less than zero. From the graph of \( f(x) = 7 - 5 \log{x} \), we can visually determine these regions. Let's break it down:

• First, locate the x-intercept at \( x = 10^{\frac{7}{5}} \), where the function is zero.
• For \( f(x) < 0 \), observe the area where the function is below the x-axis. As the solution indicates, this occurs when \( x > 10^{\frac{7}{5}} \).
• Conversely, \( f(x) \geq 0 \) is found where the function is on or above the x-axis. This inequality holds when \( x \leq 10^{\frac{7}{5}} \) and \( x > 0 \).

Understanding inequalities through graphs simplifies identifying the solution sets, making it less abstract and more intuitive. Graphical regions thus correspond to solutions, allowing you to confirm or find solutions without delving into complex algebraic manipulations.

Exponential Functions

Exponential functions and logarithms are interconnected; understanding one requires knowledge of the other. When solving the equation \( 7 - 5 \log{x} = 0 \), converting it into an exponential form through the logarithmic relation \( \log{x} = \frac{7}{5} \) gives:\[ x = 10^{\frac{7}{5}} \]This step transforms the equation from logarithmic to exponential, making it solvable by straightforward calculation. Key points to remember include:

• Exponential expressions like \( 10^{\frac{7}{5}} \) exemplify how changing the base of logarithms results in expressions easy to compute using calculators or software.
• This approach showcases the power of exponential functions to express solutions involving large or small numbers in a concise, manageable form.

Recognizing the link between logarithmic and exponential functions helps to solve complex equations and gain a deeper understanding of their applicability, especially when transitioning between different mathematical representations.