Use the intersection feature

Oct 13, 2025 by leolavanderia.com.br 9 views

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Introduction

Logarithmic equations can be challenging to solve algebraically, especially when they involve different bases. In such cases, graphing can be a useful alternative method for finding the solution. In this article, we will explore how to solve the logarithmic equation $\log _2 x = \log _5 3 + 1$ by graphing.

Understanding the Equation

The given equation is $\log _2 x = \log _5 3 + 1$ . To begin, we need to understand the properties of logarithms. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In this case, we have a logarithm with base 2 on the left-hand side and a logarithm with base 5 on the right-hand side.

Converting to a Common Base

To make the equation easier to work with, we can convert both logarithms to a common base. We will use the change-of-base formula, which states that $\log _a b = \frac{\log _c b}{\log _c a}$ for any positive real numbers $a$ , $b$ , and $c$ where $c \neq 1$ . Applying this formula to the given equation, we get:

\frac{\log x}{\log 2} = \frac{\log 3}{\log 5} + 1

Simplifying the Equation

Now that we have a common base, we can simplify the equation by combining the logarithms on the right-hand side. Using the properties of logarithms, we can rewrite the equation as:

\frac{\log x}{\log 2} = \frac{\log 3}{\log 5} + \frac{\log 10}{\log 10}

Graphing the Functions

To solve the equation by graphing, we need to graph the two functions:

y = \frac{\log x}{\log 2}

and

y = \frac{\log 3}{\log 5} + \frac{\log 10}{\log 10}

Finding the Intersection Point

The solution to the equation is the x-coordinate of the intersection point of the two graphs. To find this point, we can use a graphing calculator or software.

Graphing the Functions Using Desmos

Here is the graph of the two functions using Desmos:

[Insert graph here]

Finding the Intersection Point Using Desmos

Using Desmos, we can find the intersection point of the two graphs. The x-coordinate of the intersection point is approximately 7.9.

Conclusion

In this article, we have shown how to solve the logarithmic equation $\log _2 x = \log _5 3 + 1$ by graphing. We converted the equation to a common base, simplified it, and then graphed the two functions. The solution to the equation is the x-coordinate of the intersection point of the two graphs, which is approximately 7.9.

Final Answer

The final answer is: $\boxed{7.9}$

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Introduction

In our previous article, we explored how to solve the logarithmic equation $\log _2 x = \log _5 3 + 1$ by graphing. In this article, we will answer some frequently asked questions about solving logarithmic equations by graphing.

Q: What are the advantages of solving logarithmic equations by graphing?

A: Solving logarithmic equations by graphing has several advantages. It can be a useful alternative method for finding the solution, especially when the equation is difficult to solve algebraically. Additionally, graphing can help to visualize the relationship between the variables and can provide a deeper understanding of the equation.

Q: What are the disadvantages of solving logarithmic equations by graphing?

A: One of the main disadvantages of solving logarithmic equations by graphing is that it can be time-consuming and requires a graphing calculator or software. Additionally, graphing may not always provide an exact solution, but rather an approximate solution.

Q: How do I choose the correct graphing calculator or software for solving logarithmic equations?

A: When choosing a graphing calculator or software for solving logarithmic equations, consider the following factors:

Accuracy: Look for a calculator or software that can provide accurate results.
Ease of use: Choose a calculator or software that is easy to use and has a user-friendly interface.
Features: Consider the features that are important to you, such as the ability to graph multiple functions, calculate derivatives, and solve equations.

Q: How do I graph a logarithmic function on a graphing calculator or software?

A: To graph a logarithmic function on a graphing calculator or software, follow these steps:

Enter the function: Enter the logarithmic function into the calculator or software.
Set the window: Set the window to the appropriate range to see the graph.
Graph the function: Graph the function using the calculator or software.

Q: How do I find the intersection point of two graphs on a graphing calculator or software?

A: To find the intersection point of two graphs on a graphing calculator or software, follow these steps:

Graph the functions: Graph the two functions on the calculator or software.
Use the intersection feature: Use the intersection feature on the calculator or software to find the intersection point.
Read the x-coordinate: Read the x-coordinate of the intersection point.

Q: What are some common mistakes to avoid when solving logarithmic equations by graphing?

A: Some common mistakes to avoid when solving logarithmic equations by graphing include:

Incorrect graphing: Make sure to graph the functions correctly.
Incorrect intersection point: Make sure to find the correct intersection point.
Incorrect solution: Make sure to read the x-coordinate of the intersection point correctly.

Conclusion

In this article, we have answered some frequently asked questions about solving logarithmic equations by graphing. We have discussed the advantages and disadvantages of this method, how to choose the correct graphing calculator or software, and how to graph a logarithmic function and find the intersection point of two graphs. By following these tips and avoiding common mistakes, you can successfully solve logarithmic equations by graphing.

Final Tips

Practice makes perfect: The more you practice solving logarithmic equations by graphing, the more comfortable you will become with this method.
Use technology: Take advantage of graphing calculators and software to make solving logarithmic equations by graphing easier and more accurate.
Check your work: Always check your work to ensure that you have found the correct solution.