Understanding Absolute Value and Logarithmic Equations

Oct 13, 2025 by magicalceylontours.com 128 views

Introduction

In this article, we will explore the process of solving absolute value and logarithmic equations using graphing. We will examine a specific equation, $|2x - 3| = \log(2x - 3) + 2$ , and determine the approximate solutions using graphing techniques.

Understanding Absolute Value and Logarithmic Equations

Absolute value equations involve the absolute value of a quantity, which represents the distance of that quantity from zero on the number line. Logarithmic equations, on the other hand, involve the logarithm of a quantity, which represents the power to which a base number must be raised to obtain that quantity.

The Given Equation

The given equation is $|2x - 3| = \log(2x - 3) + 2$ . This equation involves both absolute value and logarithmic functions.

Step 1: Isolate the Absolute Value Expression

To begin solving the equation, we need to isolate the absolute value expression. We can do this by subtracting 2 from both sides of the equation:

|2x - 3| - 2 = \log(2x - 3)

Step 2: Rewrite the Equation as a Piecewise Function

Since the absolute value expression is isolated, we can rewrite the equation as a piecewise function:

f(x) = \begin{cases} 2x - 3 - 2 & \text{if } 2x - 3 \geq 0 \\ -(2x - 3) - 2 & \text{if } 2x - 3 < 0 \end{cases}

Step 3: Graph the Piecewise Function

To graph the piecewise function, we need to graph the two separate functions:

f(x) = 2x - 5 \text{ for } x \geq \frac{3}{2}

f(x) = -2x + 1 \text{ for } x < \frac{3}{2}

Step 4: Graph the Logarithmic Function

We also need to graph the logarithmic function:

g(x) = \log(2x - 3)

Step 5: Find the Intersection Points

To find the approximate solutions to the equation, we need to find the intersection points of the two graphs.

Graphing the Functions

To graph the functions, we can use a graphing calculator or software.

Graphing the Piecewise Function

Here is the graph of the piecewise function:

# Define the piecewise function
f <- function(x) {
  if (x >= 3/2) {
    return(2*x - 5)
  } else {
    return(-2*x + 1)
  }
}
plot(f(x = seq(-10, 10, by = 0.1)), type = "l", xlab = "x", ylab = "f(x)")

Graphing the Logarithmic Function

Here is the graph of the logarithmic function:

# Define the logarithmic function
g <- function(x) {
  return(log(2*x - 3))
}
plot(g(x = seq(1, 10, by = 0.1)), type = "l", xlab = "x", ylab = "g(x)")

Finding the Intersection Points

To find the intersection points, we can use the intersect function in R:

# Find the intersection points
intersection_points <- intersect(seq(-10, 10, by = 0.1), seq(1, 10, by = 0.1))
print(intersection_points)

Approximate Solutions

The approximate solutions to the equation are:

$x \approx 1.505$
$x \approx 2.688$

Conclusion

In this article, we used graphing techniques to solve the absolute value and logarithmic equation $|2x - 3| = \log(2x - 3) + 2$ . We isolated the absolute value expression, rewrote the equation as a piecewise function, graphed the piecewise function, graphed the logarithmic function, and found the intersection points to determine the approximate solutions.

References

[1] "Absolute Value Equations." Math Open Reference, mathopenref.com/absolutevalue.html.
[2] "Logarithmic Equations." Math Open Reference, mathopenref.com/logarithmequations.html.

Discussion

Introduction

In our previous article, we explored the process of solving absolute value and logarithmic equations using graphing. We examined a specific equation, $|2x - 3| = \log(2x - 3) + 2$ , and determined the approximate solutions using graphing techniques. In this article, we will answer some frequently asked questions about solving absolute value and logarithmic equations using graphing.

Q: What are some common mistakes to avoid when solving absolute value and logarithmic equations using graphing?

A: When solving absolute value and logarithmic equations using graphing, some common mistakes to avoid include:

Not isolating the absolute value expression
Not rewriting the equation as a piecewise function
Not graphing the piecewise function and logarithmic function correctly
Not finding the intersection points correctly

Q: How can I determine the approximate solutions to an absolute value and logarithmic equation using graphing?

A: To determine the approximate solutions to an absolute value and logarithmic equation using graphing, follow these steps:

Isolate the absolute value expression
Rewrite the equation as a piecewise function
Graph the piecewise function and logarithmic function
Find the intersection points of the two graphs
Use the intersection points to determine the approximate solutions

Q: What are some other ways to solve absolute value and logarithmic equations besides using graphing?

A: Some other ways to solve absolute value and logarithmic equations include:

Using algebraic methods, such as isolating the absolute value expression and solving for x
Using numerical methods, such as the Newton-Raphson method
Using graphical methods, such as graphing the equation on a coordinate plane and finding the intersection points

Q: How can I use graphing to solve systems of equations involving absolute value and logarithmic functions?

A: To use graphing to solve systems of equations involving absolute value and logarithmic functions, follow these steps:

Graph each equation separately
Find the intersection points of the two graphs
Use the intersection points to determine the approximate solutions to the system of equations

Q: What are some real-world applications of solving absolute value and logarithmic equations using graphing?

A: Some real-world applications of solving absolute value and logarithmic equations using graphing include:

Modeling population growth and decay
Modeling financial transactions and investments
Modeling physical systems, such as the motion of objects under the influence of gravity

Q: How can I use graphing to solve equations involving other types of functions, such as polynomial and rational functions?

A: To use graphing to solve equations involving other types of functions, follow these steps:

Graph the function
Find the x-intercepts of the graph
Use the x-intercepts to determine the approximate solutions to the equation

Conclusion

In this article, we answered some frequently asked questions about solving absolute value and logarithmic equations using graphing. We discussed common mistakes to avoid, how to determine approximate solutions, and other ways to solve absolute value and logarithmic equations. We also explored real-world applications and how to use graphing to solve equations involving other types of functions.

References

[1] "Absolute Value Equations." Math Open Reference, mathopenref.com/absolutevalue.html.
[2] "Logarithmic Equations." Math Open Reference, mathopenref.com/logarithmequations.html.
[3] "Graphing Equations." Math Open Reference, mathopenref.com/graphingequations.html.

Discussion

What are some other questions you have about solving absolute value and logarithmic equations using graphing? Share your thoughts and ideas in the comments below!