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Chapter 5: Problem 99

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 \(f(x) \geq 0\) $$f(x)=8-4 \log _{5} x$$

Short Answer

Expert verified
\(f(x) = 0\) at \(x = 25\); \(f(x) < 0\) when \(x > 25\); \(f(x) \geq 0\) when \(x \leq 25\).

Step by step solution

01

Set up the equation

Begin by setting the function equal to zero to solve for when \(f(x) = 0\). We'll solve the equation \[8 - 4 \log_{5}(x) = 0\].
02

Isolate the logarithmic term

Subtract 8 from both sides of the equation to isolate the logarithmic term: \[-4 \log_{5}(x) = -8\].
03

Simplify the equation

Divide both sides by -4 to further simplify:\[\log_{5}(x) = 2\].
04

Solve the logarithmic equation

Rewrite the logarithmic equation in exponential form to solve for \(x\):\[x = 5^2\].
05

Calculate the value

Calculate \(5^2\) to find the value of \(x\):\[x = 25\].
06

Graphical Interpretation for Inequalities

Using the graph of \(y = f(x) = 8 - 4 \log_{5}(x)\), identify where \(f(x) < 0\) and \(f(x) \geq 0\). The critical point where the graph intersects the x-axis corresponds to \(x = 25\).
07

Determine intervals from the graph

From the graph:- \(f(x) < 0\) when \(x > 25\).- \(f(x) \geq 0\) when \(x \leq 25\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving equations is a fundamental skill in algebra and calculus, involving finding the unknown values that satisfy a given mathematical statement. Here, we examine how to solve the equation \( f(x) = 0 \).

Given the function \( f(x) = 8 - 4 \log_5{x} \), to solve \( 8 - 4 \log_5{x} = 0 \):
  • Start by isolating the logarithmic expression by subtracting 8 from both sides, leading to \( -4 \log_5{x} = -8 \).
  • Next, dividing both sides by -4 provides \( \log_5{x} = 2 \).
  • Recognize that the equation \( \log_5{x} = 2 \) can be rewritten in exponential form: \( x = 5^2 \).
  • Finally, calculating \( 5^2 \) reveals that \( x = 25 \) is the solution to the equation.
This process involves understanding both algebraic manipulation and properties of logarithms, particularly the ability to switch between logarithmic and exponential forms.
Inequalities
Inequalities describe a range of values that a function can take, rather than pinpointing specific points. Once you solve an equation, interpreting these solutions as inequalities gives insight into how the function behaves across different values of \( x \).

For the function \( f(x) = 8 - 4 \log_5{x} \), determine where it satisfies \( f(x) < 0 \) and \( f(x) \geq 0 \):
  • The solution \( x = 25 \) divides the number line into intervals. This point is where the function crosses the x-axis.
  • Analyze the behavior of \( f(x) \) around this point using its inequality:
    • For \( x > 25 \), substitute a value like \( x = 30 \) into \( f(x) \) to see it results in a negative value, showing \( f(x) < 0 \).
    • For \( x \leq 25 \), substituting values like \( x = 20 \), the resulting value is positive or zero, indicating \( f(x) \geq 0 \).
By determining these intervals, one can understand where the function outputs are positive and where they turn negative.
Graphical Interpretation
Graphical interpretation provides a visual perspective on both solutions of equations and the nature of inequalities. By examining the graph of a function, you can identify critical points and assess changes in sign.

For the function \( y = f(x) = 8 - 4 \log_5{x} \):
  • The graph helps confirm the exact point where \( f(x) = 0 \), known as the x-intercept, which we solved to be \( x = 25 \).
  • Observing the graph provides immediate insight into the behavior of the inequality.
    It illustrates visually where the curve dips below the x-axis \( (x > 25) \) and where it stays on or above it \( (x \leq 25) \).
This graphical method supports the analytical solution by offering a more tangible understanding of the function's behavior. It is particularly beneficial for quickly assessing inequality solutions, clarifying interval domains without complex calculations.

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