Question

In Problems \(73-86,\) use a graphing utility to solve each equation. Express your answer rounded to two decimal places. $$ \log _{5}(x+1)-\log _{4}(x-2)=1 $$

Short answer

The solution to the equation \( \log_5(x+1) - \log_4(x-2) = 1 \) is \ x \ approximately equal to 2.94.

Step-by-step solution

- Understand the Problem

The given equation is \( \log_5(x+1) - \log_4(x-2) = 1 \). The goal is to find the value of \ x \ rounded to two decimal places using a graphing utility.

- Simplify the Equation

Rewrite the equation using the properties of logarithms. Since the logarithms have different bases, express the logarithms in terms of a common base or directly use a graphing utility: \[ \log_5(x+1) = 1 + \log_4(x-2) \].

- Convert to Exponential Form

Rewrite both logarithmic functions in their exponential forms: \[ x + 1 = 5^{(\text{right})} \] and \[ x - 2 = 4^{(\text{left})} \], where 'right' and 'left' correspond to their initial placement in the problem.

- Graph the Functions

Use a graphing calculator or software to graph \[ y = \log_5(x+1) \] and \[ y = 1 + \log_4(x-2) \] on the same coordinate plane. Observe the point of intersection.

- Locate the Intersection Point

Find the x-coordinate of the intersection point. This coordinate is the solution to the equation. Use the graphing utility's intersection function to find the precise value.

- Round to Two Decimal Places

Round the x-coordinate of the intersection point to two decimal places.

Key concepts

logarithmic equations

Logarithmic equations are equations that include logarithms with variables in their arguments. These types of equations often require manipulation and an understanding of logarithm properties to solve. In our example, we have the equation: \( \log_5(x+1) - \log_4(x-2) = 1 \).

Understanding how logarithms work is crucial. The basic properties include:

• The product rule: \log_b(mn) = \log_b(m) + \log_b(n) \
• The quotient rule: \log_b(m/n) = \log_b(m) - \log_b(n) \
• The power rule: \log_b(m^n) = n\log_b(m) \

For this specific problem, different bases are involved, which complicates direct manipulation. This makes using a graphing utility an effective strategy to find the solution by visual means.

graphing utility

A graphing utility, such as a graphing calculator or software, helps in solving equations by visualizing them. In our problem, we use the graphing utility to graph the equations \( y = \log_5(x+1) \) and \( y =1 + \log_4(x-2) \).

Here’s how to proceed:

• Input the function \( y = \log_5(x+1) \) into the graphing utility.
• Next, input the function \( y = 1+ \log_4(x-2) \).
• Observe where the two graphs intersect on the coordinate plane.

Graphing utilities make it easier to handle equations with different bases, as in this example. You can locate and zoom into the intersection point, which represents the solution to the equation. These tools often have built-in features for finding intersections, enabling more precise answers.

intersection point

In solving equations graphically, the intersection point of the graphs is crucial. To solve the given logarithmic equation by graphing, you need to identify where \( y = \log_5(x+1) \) and \( y = 1 + \log_4(x-2) \) meet.

Here’s a detailed process:

• Graph both functions on the same coordinate plane using your graphing utility.
• Look for the point where the two graphs intersect.
• Use the graphing utility's intersection feature to determine the x-coordinate of this point.

The x-coordinate at the intersection point is the solution to the original equation. Round this x-value to two decimal places to meet the problem's requirements.