Solve For $x$.$\log _3(x+25) - \log _3(x-1) = 3$
Introduction to Logarithmic Equations
Logarithmic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science. In this article, we will focus on solving a specific logarithmic equation, which involves the use of logarithmic properties and algebraic manipulations.
Understanding the Given Equation
The given equation is $\log _3(x+25) - \log _3(x-1) = 3$. This equation involves logarithms with base 3, and we need to solve for the variable $x$. The equation can be rewritten as $\log _3\left(\frac{x+25}{x-1}\right) = 3$, using the property of logarithms that states $\log _a(b) - \log _a(c) = \log _a\left(\frac{b}{c}\right)$.
Applying Logarithmic Properties
To solve the equation, we can start by applying the property of logarithms that states $\log _a(b) = c$ is equivalent to $a^c = b$. In this case, we have $\log _3\left(\frac{x+25}{x-1}\right) = 3$, which is equivalent to $3^3 = \frac{x+25}{x-1}$.
Simplifying the Equation
We can simplify the equation by evaluating the left-hand side, which is $3^3 = 27$. Therefore, we have $27 = \frac{x+25}{x-1}$.
Cross-Multiplying
To eliminate the fraction, we can cross-multiply the equation, which gives us $27(x-1) = x+25$.
Expanding and Simplifying
We can expand the left-hand side of the equation by multiplying 27 with the terms inside the parentheses, which gives us $27x - 27 = x + 25$. We can then simplify the equation by combining like terms, which gives us $26x - 27 = 25$.
Isolating the Variable
To isolate the variable $x$, we can add 27 to both sides of the equation, which gives us $26x = 52$. We can then divide both sides of the equation by 26, which gives us $x = 2$.
Conclusion
In this article, we have solved a logarithmic equation involving the use of logarithmic properties and algebraic manipulations. We have shown that the solution to the equation is $x = 2$. This result can be verified by plugging the value of $x$ back into the original equation and checking that it satisfies the equation.
Final Answer
The final answer to the equation $\log _3(x+25) - \log _3(x-1) = 3$ is $x = 2$.
Additional Tips and Tricks
- When solving logarithmic equations, it is essential to apply the properties of logarithms to simplify the equation.
- Cross-multiplying can be a useful technique to eliminate fractions and simplify the equation.
- Be careful when expanding and simplifying the equation, as it is easy to make mistakes.
- Always verify the solution by plugging the value of $x$ back into the original equation.
Common Mistakes to Avoid
- Failing to apply the properties of logarithms can lead to incorrect solutions.
- Not cross-multiplying can result in a more complex equation that is difficult to solve.
- Making mistakes when expanding and simplifying the equation can lead to incorrect solutions.
- Not verifying the solution can result in an incorrect answer.
Real-World Applications
Logarithmic equations have numerous real-world applications in various fields such as physics, engineering, and computer science. Some examples include:
- Modeling population growth and decay
- Analyzing financial data and predicting stock prices
- Solving problems involving exponential decay and growth
- Modeling chemical reactions and kinetics
Conclusion
In conclusion, solving logarithmic equations requires a deep understanding of logarithmic properties and algebraic manipulations. By applying the properties of logarithms and using techniques such as cross-multiplying and expanding, we can solve complex logarithmic equations and obtain accurate solutions.
Introduction
In the previous article, we solved a logarithmic equation involving the use of logarithmic properties and algebraic manipulations. In this article, we will provide a Q&A section to address common questions and concerns related to logarithmic equations.
Q: What is a logarithmic equation?
A: A logarithmic equation is an equation that involves logarithms, which are the inverse of exponential functions. Logarithmic equations can be used to model real-world problems involving exponential growth and decay.
Q: How do I solve a logarithmic equation?
A: To solve a logarithmic equation, you need to apply the properties of logarithms, such as the product rule, quotient rule, and power rule. You can also use algebraic manipulations, such as cross-multiplying and expanding, to simplify the equation.
Q: What is the difference between a logarithmic equation and an exponential equation?
A: A logarithmic equation involves logarithms, while an exponential equation involves exponential functions. Logarithmic equations can be used to model problems involving exponential growth and decay, while exponential equations can be used to model problems involving exponential functions.
Q: How do I apply the properties of logarithms?
A: To apply the properties of logarithms, you need to understand the following rules:
- Product rule:
- Quotient rule:
- Power rule:
Q: What is the significance of the base of a logarithm?
A: The base of a logarithm is an important concept in logarithmic equations. The base of a logarithm determines the type of logarithm, such as natural logarithm (base e) or common logarithm (base 10).
Q: How do I choose the correct base for a logarithmic equation?
A: To choose the correct base for a logarithmic equation, you need to consider the problem and the units involved. For example, if you are working with financial data, you may want to use a base of 10 (common logarithm) or e (natural logarithm).
Q: What are some common mistakes to avoid when solving logarithmic equations?
A: Some common mistakes to avoid when solving logarithmic equations include:
- Failing to apply the properties of logarithms
- Not cross-multiplying
- Making mistakes when expanding and simplifying the equation
- Not verifying the solution
Q: How do I verify the solution to a logarithmic equation?
A: To verify the solution to a logarithmic equation, you need to plug the value of the variable back into the original equation and check that it satisfies the equation.
Q: What are some real-world applications of logarithmic equations?
A: Logarithmic equations have numerous real-world applications in various fields such as physics, engineering, and computer science. Some examples include:
- Modeling population growth and decay
- Analyzing financial data and predicting stock prices
- Solving problems involving exponential decay and growth
- Modeling chemical reactions and kinetics
Q: Can logarithmic equations be used to model non-linear relationships?
A: Yes, logarithmic equations can be used to model non-linear relationships. Logarithmic equations can be used to model problems involving exponential growth and decay, which can be non-linear.
Q: How do I use logarithmic equations to model real-world problems?
A: To use logarithmic equations to model real-world problems, you need to:
- Identify the problem and the units involved
- Choose the correct base for the logarithm
- Apply the properties of logarithms
- Use algebraic manipulations to simplify the equation
- Verify the solution
Conclusion
In conclusion, logarithmic equations are a powerful tool for modeling real-world problems involving exponential growth and decay. By understanding the properties of logarithms and applying algebraic manipulations, you can solve complex logarithmic equations and obtain accurate solutions.