Problem 56 - College Algebra

Question

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. $$ -7+\log _{3}(4-x)=-6 $$

Step-by-Step Solution

Answer

x = 1; Verified by graphing.

Step 1: Isolate the Logarithmic Expression

We start with the equation \[-7 + \log_{3}(4 - x) = -6\]We need to isolate the logarithmic term, which is \(\log_{3}(4 - x)\). We do this by adding 7 to both sides:\[\log_{3}(4 - x) = -6 + 7\]\[\log_{3}(4 - x) = 1\].

Step 2: Rewrite the Logarithmic Equation in Exponential Form

The equation \(\log_{3}(4 - x) = 1\) can be rewritten in exponential form. This translates as:\[4 - x = 3^{1}\]Since \(3^{1} = 3\), the equation becomes:\[4 - x = 3\].

Step 3: Solve for x

To solve for \(x\), subtract 4 from both sides:\[-x = 3 - 4\]This simplifies to:\[-x = -1\].Multiply through by -1 to get:\[x = 1\].

Step 4: Verification by Graphing

Graph the expressions on both sides of the original equation: \(-7 + \log_{3}(4 - x)\) and \(-6\).- The graph of \(-7 + \log_{3}(4 - x)\) will intersect the horizontal line \(y = -6\) at \(x = 1\).- This confirms that the point of intersection verifies our solution \(x=1\).

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve a logarithm with an unknown variable. In these equations, the variable frequently resides inside the logarithmic expression. Understanding how to manipulate logarithmic equations is crucial for solving them effectively. Remember, a logarithm tells us the power to which a number (the base of the logarithm) must be raised to obtain another number. Thus, solving a logarithmic equation often involves rewriting it in its exponential form.How to Solve a Logarithmic Equation:

• Isolate the logarithmic part of the equation if needed.
• Once isolated, use the properties of logarithms to rewrite the equation in exponential form. This is because if \(\log_b(a) = c\), then \(b^c = a\).
• Solve the resulting exponential equation for the unknown variable.

In the given exercise, \(-7 + \log_{3}(4 - x) = -6\), first isolate the logarithmic expression. This becomes \(\log_{3}(4 - x) = 1\), which, when rewritten in exponential form, is \(4-x = 3^1\). Solving this leads us to \(x=1\). This method clearly shows how using the principles of logarithms and exponential functions helps in finding the solution for the variable.

Graphical Verification

Graphical verification is the process of using a graph to confirm the solution of an equation. This method is incredibly useful for visual learners and provides a visual confirmation that can supplement algebraic solutions. To perform graphical verification:

• Plot both sides of the given equation as separate functions on a graph.
• Observe where these two graphs intersect. The x-value of this point is the solution to the equation.

For example, in the exercise, we have the original equation \(-7 + \log_{3}(4 - x) = -6\). Graph these separately:- The graph of \(-7 + \log_{3}(4 - x)\).- The graph of the constant function \(y = -6\).The intersection point where these two graphs meet indicates the solution to the equation. For this exercise, the intersection confirms that the solution is \(x = 1\). This is a compelling way to verify that our algebraic solution is correct and builds confidence in the method.

Intersection Points

Intersection points on a graph are where two functions meet, and they represent the solutions to an equation when graphed. Understanding how intersection points verify solutions is crucial in graphically verifying an equation's solution.Here's what you need to know about intersection points:

• The x-coordinate of the intersection denotes the solution for the variable x.
• If graphs intersect at a point, it confirms the accuracy of the algebraically derived solution.
• If there's no intersection, this could indicate that there's no real solution.

In the presented exercise, the intersection is between \(-7 + \log_{3}(4 - x)\) and \(-6\). The graph shows that these functions intersect at \(x = 1\). This means at \(x = 1\), both equations yield the same result, thereby confirming the solution is correct. Using intersection points helps visually substantiate the problem-solving process, making abstract equations easier to understand.