Problem 54 - 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. \(\ln (x-5)=1\)

Step-by-Step Solution

Answer

The solution is \( x \approx 7.718 \).

Step 1: Identify the equation

The given equation is \( \ln(x-5) = 1 \). This is a logarithmic equation, where you need to solve for \( x \).

Step 2: Rewrite in exponential form

To solve the logarithmic equation \( \ln(x-5) = 1 \), rewrite it in its exponential form. Recall that \( \ln(a) = b \) translates to \( a = e^b \). Hence, \( x-5 = e^1 \).

Step 3: Simplify and solve for \(x\)

Now, solve for \( x \) by isolating it. So, \( x - 5 = e \) implies \( x = e + 5 \).

Step 4: Calculate numerical value

Calculate \( e + 5 \) using the approximate value of \( e \) which is about 2.718. So, \( x \approx 2.718 + 5 = 7.718 \).

Step 5: Graph both sides of the equation

Graph the functions \( f(x) = \ln(x-5) \) and \( g(x) = 1 \) on the same coordinate system. The intersection point of these graphs, if it exists, corresponds to the solution of the equation.

Step 6: Verify the solution

Observe the graph where the line \( y = 1 \) intersects the curve \( \ln(x-5) \). They should intersect at the point where \( x \approx 7.718 \), confirming that this is the solution.

Key Concepts

Exponential Form

In mathematics, converting a logarithmic equation to its exponential form is an important step to solve for unknown variables. The equation given is \( \ln(x-5) = 1 \). Here, \( \ln \) stands for the natural logarithm, and it indicates the base \( e \), which is an irrational constant approximately equal to 2.718.
To rewrite \( \ln(a) = b \) in its exponential form, you express it as \( a = e^b \). This transformation makes it easier to solve for \( x \) when dealing with logarithmic equations. For our specific problem, rewriting \( \ln(x-5) = 1 \) in exponential form results in \( x-5 = e^1 \).
This means \( x-5 \, \) is equal to \( \, e \), helping us isolate \( x \) and find its value by performing algebraic operations.

Graphing Equations

Graphing mathematical equations provides a visual representation of solutions and relationships. When we graph equations like \( f(x) = \ln(x-5) \) and \( g(x) = 1 \), we can observe how different functions behave and where they meet.
In the provided exercise, plotting these functions on the same coordinate axes can help you visually confirm the solution through their intersection. The curve \( \ln(x-5) \) will increase and pass through a point where the value of \( x \) makes the equation \( \ln(x-5) = 1 \) true.
Using graphing tools or plotting by hand:

• Mark key points where \( f(x) \) is defined, starting at \( x > 5 \) since \( \ln(x-5) \) is not defined for values \( x \leq 5 \).
• Draw the horizontal line for \( y = 1 \).
• The intersection where both graphs meet gives the value \( x \) computed earlier.

This approach often clarifies theoretical solutions through practical visualization.

Point of Intersection

The point of intersection between two graphs is a key concept that visually confirms mathematical solutions. When solving equations graphically, the intersection offers an intuitive way to verify calculations. In the problem \( \ln(x-5) = 1 \), finding the intersection point involves the graphs of \( f(x) = \ln(x-5) \) and \( g(x) = 1 \).
At the intersection point:

• Both functions yield the same output value for a specific \( x \).
• This common output, 1, corresponds to the respective input, giving the solution for \( x \).

The solution \( x \approx 7.718 \) is effectively confirmed by observing where the curve and line meet. This method not only supports the accuracy of computations but also enhances understanding by connecting algebraic results to visual interpretations.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with base \( e \). The constant \( e \) is approximately 2.718 and serves as a fundamental element in calculus and exponential growth models.
In the expression \( \ln(x-5) = 1 \), the natural logarithm transforms multiplicative relationships into additive ones, simplifying the process of solving exponential equations. It helps translate between linear and exponential forms, providing insights scale through a natural growth rate.
Understanding natural logarithms involves:

• Recognizing \( \ln(a) = b \) can be rewritten as \( a = e^b \).
• Interpreting the logarithm as the power to which \( e \) must be raised to produce \( a \).

This mathematical tool is crucial in solving equations where natural exponential bases are involved, as it enables deeper comprehension of growth dynamics and decay processes in real-world contexts.