Solve The Equation For { N $} : : : { \log _9(n+7)=4 \}

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific logarithmic equation involving a base of 9. The equation is $\log_9(n+7) = 4$. Our goal is to isolate the variable $n$ and find its value.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's take a moment to understand what logarithmic equations are. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a(c) = b$. Logarithmic equations can be solved using various techniques, including the use of logarithmic properties and the change of base formula.

Solving the Equation

Now that we have a basic understanding of logarithmic equations, let's focus on solving the equation $\log_9(n+7) = 4$. To solve this equation, we can use the definition of a logarithm, which states that if $\log_a(b) = c$, then $a^c = b$. Applying this definition to our equation, we get:

$$9^4 = n + 7$$

Using Exponentiation to Simplify the Equation

Now that we have the equation in exponential form, we can simplify it by evaluating the left-hand side. Since $9^4 = 6561$, we can rewrite the equation as:

$$6561 = n + 7$$

Isolating the Variable $n$

To isolate the variable $n$, we need to get rid of the constant term on the right-hand side. We can do this by subtracting 7 from both sides of the equation:

$$6561 - 7 = n$$

Simplifying the Equation

Now that we have the equation in a simpler form, we can evaluate the left-hand side to find the value of $n$. Since $6561 - 7 = 6554$, we can rewrite the equation as:

$$n = 6554$$

Conclusion

In this article, we solved a logarithmic equation involving a base of 9. We used the definition of a logarithm to rewrite the equation in exponential form, and then simplified it by evaluating the left-hand side. Finally, we isolated the variable $n$ by subtracting 7 from both sides of the equation. The value of $n$ is 6554.

Additional Tips and Tricks

When solving logarithmic equations, it's essential to remember the following tips and tricks:

• Use the definition of a logarithm to rewrite the equation in exponential form.
• Simplify the equation by evaluating the left-hand side.
• Isolate the variable by adding or subtracting the same value from both sides of the equation.
• Use logarithmic properties and the change of base formula to solve more complex logarithmic equations.

Common Mistakes to Avoid

When solving logarithmic equations, it's easy to make mistakes. Here are some common mistakes to avoid:

• Forgetting to use the definition of a logarithm to rewrite the equation in exponential form.
• Not simplifying the equation by evaluating the left-hand side.
• Not isolating the variable by adding or subtracting the same value from both sides of the equation.
• Using the wrong logarithmic properties or the change of base formula.

Real-World Applications

Logarithmic equations have numerous real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design and optimize systems.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a(c) = b$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the definition of a logarithm to rewrite the equation in exponential form. Then, simplify the equation by evaluating the left-hand side and isolate the variable by adding or subtracting the same value from both sides of the equation.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, $\log_9(n+7) = 4$ is a logarithmic equation, while $9^4 = n + 7$ is an exponential equation.

Q: How do I use the change of base formula to solve a logarithmic equation?

A: The change of base formula is a formula that allows you to change the base of a logarithm. It is given by:

$$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$$

where $a$, $b$, and $c$ are positive real numbers.

Q: What is the logarithmic property of addition?

A: The logarithmic property of addition states that:

$$\log_a(b) + \log_a(c) = \log_a(bc)$$

Q: What is the logarithmic property of multiplication?

A: The logarithmic property of multiplication states that:

$$\log_a(b) \cdot \log_a(c) = \log_a(bc)$$

Q: How do I use logarithmic properties to simplify a logarithmic equation?

A: To simplify a logarithmic equation using logarithmic properties, you can use the properties of addition and multiplication to combine the logarithms.

Q: What is the difference between a natural logarithm and a common logarithm?

A: A natural logarithm is a logarithm with a base of $e$, while a common logarithm is a logarithm with a base of 10.

Q: How do I convert a common logarithm to a natural logarithm?

A: To convert a common logarithm to a natural logarithm, you can use the following formula:

$$\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$$

Q: How do I convert a natural logarithm to a common logarithm?

A: To convert a natural logarithm to a common logarithm, you can use the following formula:

$$\ln(x) = \log_{10}(x) \cdot \ln(10)$$

Conclusion

In conclusion, logarithmic equations are an important topic in mathematics, and understanding how to solve them is crucial for many real-world applications. By following the steps outlined in this article, you can solve logarithmic equations with ease. Remember to use the definition of a logarithm to rewrite the equation in exponential form, simplify the equation by evaluating the left-hand side, and isolate the variable by adding or subtracting the same value from both sides of the equation. With practice and patience, you can become proficient in solving logarithmic equations and apply them to real-world problems.

Additional Resources

For more information on logarithmic equations, check out the following resources:

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

Practice Problems

Try solving the following logarithmic equations:

1. $\log_9(n+7) = 3$
2. $\log_{10}(n+5) = 2$
3. $\ln(n+3) = 1$

Answer Key

1. $n = 6561 - 7 = 6554$
2. $n = 100 - 5 = 95$
3. $n = e - 3 \approx 2.71828 - 3 = -0.28172$