Problem 104 - 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 \(f(x) \geq 0\) $$f(x)=3-2 \log _{4}(x-5)$$

Step-by-Step Solution

Answer

The solution to \( f(x) = 0 \) is \( x = 13 \). \( f(x) < 0 \) for \( x > 13 \), and \( f(x) \geq 0 \) for \( 5 < x \leq 13 \).

Step 1: Solve the Equation f(x) = 0

To solve the equation \( f(x) = 0 \), set \( 3 - 2 \log_{4}(x-5) = 0 \). First, isolate the logarithmic function: \( -2 \log_{4}(x-5) = -3 \), which simplifies to \( \log_{4}(x-5) = \frac{3}{2} \). Rewriting the equation in exponential form gives \( x - 5 = 4^{\frac{3}{2}} \). Calculate \( 4^{\frac{3}{2}} \): \( 4^{\frac{3}{2}} = (4^{1/2})^3 = 2^3 = 8 \). So, \( x - 5 = 8 \), leading to \( x = 13 \).

Step 2: Analyze the Function with a Graph

Next, consider the graph of \( y = 3 - 2 \log_{4}(x-5) \). The logarithmic function \( \log_{4}(x-5) \) is defined only for \( x > 5 \). The graph passes through the point \( x = 13 \), where \( y = 0 \). As \( x \) approaches \( 5^+ \), \( \log_{4}(x-5) \) tends towards \( -\infty \), making \( y \rightarrow \infty \). For \( x > 13 \), \( \log_{4}(x-5) > \frac{3}{2} \) making \( y < 0 \).

Step 3: Solve the Inequality f(x) < 0

For \( f(x) < 0 \), consider the region of \( x \) such that \( 3 - 2 \log_{4}(x-5) < 0 \). From the analysis, \( f(x) < 0 \) is true when \( x > 13 \) since \( \log_{4}(x-5) > \frac{3}{2} \).

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

For \( f(x) \geq 0 \), the appropriate region is where \( 3 - 2 \log_{4}(x-5) \geq 0 \). From the analysis and the equation, \( 5 < x \leq 13 \). Here, \( \log_{4}(x-5) \leq \frac{3}{2} \), satisfying the condition.

Key Concepts

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In our exercise, the function is given by \[ f(x) = 3 - 2 \log_4(x-5) \]To find where this function equals zero, we set \( f(x) = 0 \). The equation becomes:\[ 3 - 2 \log_4(x-5) = 0 \]By isolating the logarithm, we have:\[ -2 \log_4(x-5) = -3 \]which simplifies to:\[ \log_4(x-5) = \frac{3}{2} \]We convert this logarithmic equation into an exponential form:\[ x - 5 = 4^{\frac{3}{2}} \]The base 4 raised to the power of \( \frac{3}{2} \) is calculated as:\[ 4^{\frac{3}{2}} = (4^{1/2})^3 = 2^3 = 8 \]Thus, we solve for \( x \):\[ x - 5 = 8 \Rightarrow x = 13 \]This is the solution to the equation, meaning \( x = 13 \) makes the function \( f(x) \) equal zero. A clear step-by-step approach helps in understanding the logical progression to the solution.

Logarithmic Functions

Logarithmic functions involve the logarithm of a number and are the inverse operations of exponentiation. In our exercise, we deal with the function\[ 3 - 2 \log_4(x-5) \]The logarithm here is \( \log_4(x-5) \), which means "to what power must 4 be raised, to yield \( x-5 \)". Understanding this conversion is key.
The logarithmic function is defined only when its argument, \( x-5 \), is positive because you cannot take the logarithm of a non-positive number. Therefore, \( x > 5 \).

• The logarithm increases when \( x-5 \) increases.
• It becomes negative as it approaches zero when \( x \) is just a bit more than 5, causing the output of the whole function \( f(x) \) to grow large.
• The logarithmic term is greater than \( \frac{3}{2} \) when \( x > 13 \), leading to negative values for \( f(x) \).

Comprehending how logarithmic functions behave helps unravel how the equation translates into its graph. This knowledge is vital for solving inequalities related to the function.

Graphical Analysis

Graphical analysis is a visual approach to interpreting functions and solving inequalities. The given function is:\[ y = 3 - 2 \log_4(x-5) \]The graph helps see how the function behaves across different \( x \)-values, especially between critical points like where \( y = 0 \).
The logarithmic function\( \log_4(x-5) \) ensures the graph is only defined for \( x > 5 \).

• At \( x = 13 \), the function touches the \( x \)-axis, crossing from positive to negative \( y \)-values.
• As \( x \to 5^+ \), \( y \) tends toward infinity, highlighting steep early growth.
• For \( x > 13 \), the graph dips below the \( x \)-axis, signifying \( f(x) < 0 \).

To solve the inequalities:- **\( f(x) < 0 \)**: Refers to regions where the graph is below the \( x \)-axis, specifically \( x > 13 \).- **\( f(x) \geq 0 \)**: Occurs when the graph is on or above the \( x \)-axis, for \( 5 < x \leq 13 \).Breaking down the graph visually allows an immediate understanding of these intervals, connecting the algebraic and graphical solution pathways seamlessly.