Solve The Equation: $\ln (2x+3)=\ln (x+9$\]

Introduction

In mathematics, logarithmic equations are a type of equation that involves logarithmic functions. These equations can be challenging to solve, but with the right approach, they can be tackled. In this article, we will focus on solving the equation $\ln (2x+3)=\ln (x+9)$, which involves logarithmic functions. We will use various techniques to solve this equation and provide a step-by-step solution.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's understand what logarithmic equations are. A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. The logarithmic function is denoted by $\log_a(x)$, where $a$ is the base of the logarithm. The logarithmic function returns the power to which the base must be raised to produce the given number.

For example, if we have the equation $\log_2(x) = 3$, it means that $2^3 = x$, which simplifies to $x = 8$. In this example, the base of the logarithm is 2, and the result of the logarithmic function is 3.

Solving the Equation

Now that we have a basic understanding of logarithmic equations, let's focus on solving the equation $\ln (2x+3)=\ln (x+9)$. To solve this equation, we will use the property of logarithms that states that if $\log_a(x) = \log_a(y)$, then $x = y$.

Using this property, we can rewrite the equation as:

$$2x+3 = x+9$$

Simplifying the Equation

Now that we have rewritten the equation, let's simplify it by combining like terms. We can start by subtracting $x$ from both sides of the equation:

$$x+3 = 9$$

Next, we can subtract 3 from both sides of the equation:

$$x = 6$$

Verifying the Solution

To verify the solution, we can plug the value of $x$ back into the original equation:

$$\ln (2(6)+3) = \ln (6+9)$$

$$\ln (15) = \ln (15)$$

Since both sides of the equation are equal, we can conclude that the solution is correct.

Conclusion

In this article, we solved the equation $\ln (2x+3)=\ln (x+9)$ using various techniques. We started by understanding the concept of logarithmic equations and then used the property of logarithms to rewrite the equation. We then simplified the equation by combining like terms and verified the solution by plugging the value of $x$ back into the original equation. The solution to the equation is $x = 6$.

Tips and Tricks

Here are some tips and tricks to help you solve logarithmic equations:

• Use the property of logarithms that states that if $\log_a(x) = \log_a(y)$, then $x = y$.
• Simplify the equation by combining like terms.
• Verify the solution by plugging the value of $x$ back into the original equation.

Common Mistakes

Here are some common mistakes to avoid when solving logarithmic equations:

• Not using the property of logarithms to rewrite the equation.
• Not simplifying the equation by combining like terms.
• Not verifying the solution by plugging the value of $x$ back into the original equation.

Real-World Applications

Logarithmic equations have many real-world applications. Here are a few examples:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and chemical reactions.
• Engineering: Logarithmic equations are used to design electronic circuits and calculate signal strengths.

Conclusion

In conclusion, solving logarithmic equations requires a deep understanding of the concept of logarithmic functions and the properties of logarithms. By using the property of logarithms to rewrite the equation, simplifying the equation by combining like terms, and verifying the solution by plugging the value of $x$ back into the original equation, we can solve logarithmic equations with ease.

Introduction

In our previous article, we discussed how to solve the equation $\ln (2x+3)=\ln (x+9)$. In this article, we will answer some frequently asked questions about solving logarithmic equations.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. An exponential equation, on the other hand, is an equation that involves an exponential function, which is the inverse of a logarithmic function.

Q: How do I know which base to use when solving a logarithmic equation?

A: The base of the logarithm is usually given in the problem. If the base is not given, you can assume that the base is 10, unless otherwise specified.

Q: Can I use the same techniques to solve logarithmic equations with different bases?

A: Yes, you can use the same techniques to solve logarithmic equations with different bases. However, you may need to adjust the techniques slightly depending on the base.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can use the properties of logarithms to combine like terms. You can also use the fact that $\log_a(x) = \log_a(y)$ implies $x = y$.

Q: Can I use a calculator to solve logarithmic equations?

A: Yes, you can use a calculator to solve logarithmic equations. However, you should always verify the solution by plugging the value of $x$ back into the original equation.

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

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

• Not using the property of logarithms to rewrite the equation.
• Not simplifying the equation by combining like terms.
• Not verifying the solution by plugging the value of $x$ back into the original equation.

Q: How do I know if my solution is correct?

A: To verify the solution, you can plug the value of $x$ back into the original equation. If both sides of the equation are equal, then the solution is correct.

Q: Can I use logarithmic equations to model real-world problems?

A: Yes, logarithmic equations can be used to model real-world problems. For example, logarithmic equations can be used to model population growth, chemical reactions, and electronic circuits.

Q: What are some real-world applications of logarithmic equations?

A: Some real-world applications of logarithmic equations include:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and chemical reactions.
• Engineering: Logarithmic equations are used to design electronic circuits and calculate signal strengths.

Conclusion

In conclusion, solving logarithmic equations requires a deep understanding of the concept of logarithmic functions and the properties of logarithms. By using the property of logarithms to rewrite the equation, simplifying the equation by combining like terms, and verifying the solution by plugging the value of $x$ back into the original equation, we can solve logarithmic equations with ease.