Solve The Equation: $\ln (x-2) + \ln 3 = \ln (5x-7$\]

Introduction

In this article, we will delve into the world of logarithmic equations and explore a specific problem that involves the natural logarithm. The equation we will be solving is $\ln (x-2) + \ln 3 = \ln (5x-7)$. This equation may seem daunting at first, but with the right approach and techniques, we can break it down and find the solution.

Understanding Logarithmic Equations

Before we dive into the solution, it's essential to understand the basics of logarithmic equations. 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 the symbol $\log$ or $\ln$ (natural logarithm).

Properties of Logarithms

To solve logarithmic equations, we need to be familiar with the properties of logarithms. Some of the key properties include:

• Product Property: $\log (ab) = \log a + \log b$
• Quotient Property: $\log (\frac{a}{b}) = \log a - \log b$
• Power Property: $\log (a^b) = b \log a$

Solving the Equation

Now that we have a good understanding of logarithmic equations and their properties, let's dive into solving the equation $\ln (x-2) + \ln 3 = \ln (5x-7)$.

Step 1: Combine the Logarithms on the Left Side

Using the product property of logarithms, we can combine the two logarithmic terms on the left side of the equation:

$\ln (x-2) + \ln 3 = \ln (3(x-2))$

This simplifies the equation and makes it easier to work with.

Step 2: Equate the Arguments of the Logarithms

Since the logarithmic functions on both sides of the equation are equal, we can equate the arguments of the logarithms:

$3(x-2) = 5x-7$

This step is crucial in solving the equation, as it allows us to eliminate the logarithmic functions and work with a simpler equation.

Step 3: Solve for x

Now that we have a linear equation, we can solve for x:

$3x-6 = 5x-7$

Subtracting $3x$ from both sides gives us:

$-6 = 2x-7$

Adding 7 to both sides gives us:

$1 = 2x$

Dividing both sides by 2 gives us:

$\frac{1}{2} = x$

Therefore, the solution to the equation is $x = \frac{1}{2}$.

Conclusion

In this article, we solved the equation $\ln (x-2) + \ln 3 = \ln (5x-7)$ using the properties of logarithms and basic algebra. We started by combining the logarithmic terms on the left side of the equation, then equated the arguments of the logarithms, and finally solved for x. The solution to the equation is $x = \frac{1}{2}$.

Final Thoughts

Solving logarithmic equations can be challenging, but with the right approach and techniques, it's achievable. Remember to use the properties of logarithms to simplify the equation and make it easier to work with. With practice and patience, you'll become proficient in solving logarithmic equations and be able to tackle more complex problems.

Additional Resources

If you're struggling with logarithmic equations or need additional practice, here are some resources to help you:

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations Solver
• Wolfram Alpha: Logarithmic Equations Calculator

Common Mistakes to Avoid

When solving logarithmic equations, it's essential to avoid common mistakes such as:

• Forgetting to use the product property of logarithms
• Not equating the arguments of the logarithms
• Not checking the domain of the logarithmic function

Introduction

In our previous article, we solved the equation $\ln (x-2) + \ln 3 = \ln (5x-7)$ using the properties of logarithms and basic algebra. However, we know that there are many more logarithmic equations out there, and each one requires a unique approach. In this article, we'll answer some of the most frequently asked questions about 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. For example, $\log x = 2$ is a logarithmic equation, while $2^x = 8$ is an exponential equation.

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

A: When solving a logarithmic equation, you can use any base that is convenient for you. However, the most common bases are 10 (common logarithm) and e (natural logarithm). If you're given a logarithmic equation with a specific base, you should use that base when solving the equation.

Q: Can I use the product property of logarithms to combine two logarithmic terms?

A: Yes, you can use the product property of logarithms to combine two logarithmic terms. For example, $\log (ab) = \log a + \log b$. This property allows you to simplify logarithmic expressions and make them easier to work with.

Q: How do I solve a logarithmic equation with a variable in the argument?

A: To solve a logarithmic equation with a variable in the argument, you need to use the properties of logarithms to simplify the equation. For example, if you have the equation $\log (x+2) = 3$, you can use the power property of logarithms to rewrite the equation as $x+2 = 10^3$. Then, you can solve for x by subtracting 2 from both sides.

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

A: Yes, you can use a calculator to solve a logarithmic equation. However, you need to be careful when using a calculator, as it may not always give you the correct answer. It's always a good idea to check your work by plugging the solution back into the original equation.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all real numbers greater than 0. This means that the argument of the logarithmic function must be positive, or the equation will be undefined.

Q: Can I use logarithmic equations to solve exponential equations?

A: Yes, you can use logarithmic equations to solve exponential equations. For example, if you have the equation $2^x = 8$, you can take the logarithm of both sides to get $\log 2^x = \log 8$. Then, you can use the power property of logarithms to rewrite the equation as $x \log 2 = \log 8$. Finally, you can solve for x by dividing both sides by $\log 2$.

Conclusion

In this article, we answered some of the most frequently asked questions about logarithmic equations. We covered topics such as the difference between logarithmic and exponential equations, how to choose a base when solving a logarithmic equation, and how to solve logarithmic equations with variables in the argument. We also discussed the domain of a logarithmic function and how to use logarithmic equations to solve exponential equations.

Additional Resources

If you're struggling with logarithmic equations or need additional practice, here are some resources to help you:

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations Solver
• Wolfram Alpha: Logarithmic Equations Calculator

Common Mistakes to Avoid

When solving logarithmic equations, it's essential to avoid common mistakes such as:

• Forgetting to use the product property of logarithms
• Not equating the arguments of the logarithms
• Not checking the domain of the logarithmic function

By avoiding these mistakes and following the steps outlined in this article, you'll be well on your way to becoming proficient in solving logarithmic equations.