Problem 56 - College Algebra with Corequisite Support

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

The solution is \(x=1\).

Step 1: Simplify the Equation

Our given equation is \(-7+\log _{3}(4-x)=-6\). To simplify, first add 7 to both sides of the equation to isolate the logarithmic function: \(\log_{3}(4-x) = -6 + 7 = 1\).

Step 2: Rewrite in Exponential Form

Use the definition of logarithms to rewrite the equation in its exponential form. The equation \(\log_{3}(4-x) = 1\) implies that \(4-x = 3^{1}\).

Step 3: Solve for x

Now that we have \(4-x = 3\), subtract 3 from both sides: \(4-x-3=0\). This simplifies to \(1-x=0\), so add \(x\) to both sides to find that \(x=1\).

Step 4: Verify the Solution

To verify, substitute \(x=1\) back into the original equation: \(-7+\log_{3}(4-1) = -7+\log_{3}(3) = -7+1 = -6\). The solution satisfies the original equation.

Step 5: Graphical Verification

Graph the functions \(y_1 = -7 + \log_3(4-x)\) and \(y_2 = -6\) on the same set of axes. The intersection point will be at \(x=1\), confirming that the solution \(x=1\) is correct.

Key Concepts

Exponential Form

Understanding how to convert a logarithmic equation into its exponential form is crucial for solving many logarithmic problems. This transformation allows you to work with the equation by simplifying and isolating variables more easily. Consider the equation \( \log_{3}(4-x) = 1 \). The key to rewriting this in exponential form is to recall that a logarithm \( \log_{b}(a) = c \) can be expressed as \( b^{c} = a \). Therefore, for our equation, \( \log_{3}(4-x) = 1 \) translates to \( 3^{1} = 4-x \).

This exponential form helps us see that \( 4-x \) must equal 3. By writing it this way, we can solve for \( x \) more intuitively. This concept highlights the inverse relationship between logarithms and exponents and is fundamental to tackling logarithmic equations. Mastering this transformation is key to efficiently navigating through problems involving logarithms.

Graphical Verification

Graphical verification is a helpful strategy to confirm solutions to equations. By graphing both sides of the equation, you can visually identify where they intersect, which corresponds to the solution of the equation. Let's break it down with the example from the original exercise.

We have two expressions: \( y_1 = -7 + \log_3(4-x) \) and \( y_2 = -6 \). By plotting these equations on a graph:

• \( y_1 \) is a logarithmic curve that shifts vertically based on the constants.
• \( y_2 \) is a horizontal line where all points have a \(y\)-value of \(-6\).

By observing the graph, you determine the intersection point. This point of intersection represents the \(x\) value solving \(y_1 = y_2\). In this case, the intersection at \(x=1\) validates our algebraic solution. Graphical verification offers intuitive evidence, which can be particularly reassuring in complex calculations.

Solving Equations

Solving equations involves isolating the unknown variable to determine its value. In the context of our logarithmic equation \(-7+\log _{3}(4-x)=-6\), the process started with simplification. By rearranging the terms to isolate \( \log_3(4-x) \) on one side, you simplify the equation, making it more manageable to solve.

Transforming the equation into exponential form, \( 3^{1} = 4-x \), allows you to subtract 3 and ultimately solve for \( x \). Implementing each algebraic step helps resolve the equation methodically.

• While solving, ensure you perform inverse operations, such as addition or subtraction, to find the required variable \( x \).
• Remember to check the solution by substituting back into the original equation. This ensures the equation holds true, confirming accuracy.

Solving equations is about deconstructing the problem into simple, manageable steps to arrive at a precise solution.