Solve For \[$ N \$\] In The Equation:$\[ 9^n = 49 \\]

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Introduction

In this article, we will delve into solving for the variable ${ n }$ in the given exponential equation ${ 9^n = 49 }$. This equation involves a base of 9 and an exponent of ${ n }$, and we need to find the value of ${ n }$ that satisfies the equation. We will use various mathematical techniques and properties of exponents to solve for ${ n }$.

Understanding the Equation

The given equation is ${ 9^n = 49 }$. To solve for ${ n }$, we need to isolate the variable ${ n }$ on one side of the equation. However, the equation is not in a straightforward form, and we need to use some mathematical properties to simplify it.

Simplifying the Equation

We can start by simplifying the right-hand side of the equation. The number 49 can be expressed as a power of a prime number, which is 7. Specifically, ${ 49 = 7^2 }$. Therefore, we can rewrite the equation as:

${ 9^n = 7^2 }$

Using Exponent Properties

Now, we can use the property of exponents that states ${ a^m = a^n }$ if and only if ${ m = n }$. However, in this case, we have a different base on both sides of the equation. We can use the property that states ${ a^m = b^n }$ if and only if ${ a = b^{m/n} }$. Applying this property to our equation, we get:

${ 9^n = (7^2)^{1/2} }$

Simplifying the Right-Hand Side

Using the property of exponents that states ${ (a^m)^n = a^{mn} }$, we can simplify the right-hand side of the equation:

${ (7^2)^{1/2} = 7^{2 \cdot 1/2} = 7^1 = 7 }$

Equating the Bases

Now, we have:

${ 9^n = 7 }$

We can equate the bases on both sides of the equation, which gives us:

${ 9 = 7^x }$

Solving for ${ x }$

To solve for ${ x }$, we can use the property of logarithms that states ${ a^x = b }$ if and only if ${ x = \log_a b }$. Applying this property to our equation, we get:

${ x = \log_7 9 }$

Evaluating the Logarithm

To evaluate the logarithm, we can use the change of base formula, which states ${ \log_a b = \frac{\log_c b}{\log_c a} }$. Applying this formula to our equation, we get:

${ x = \frac{\log 9}{\log 7} }$

Calculating the Value

Using a calculator, we can calculate the value of ${ x }$:

${ x \approx \frac{0.95424}{0.845098} }$

${ x \approx 1.130 }$

Rounding the Value

Since ${ n }$ must be an integer, we round the value of ${ x }$ to the nearest integer:

${ n \approx 1 }$

Conclusion

In this article, we solved for the variable ${ n }$ in the equation ${ 9^n = 49 }$. We used various mathematical techniques and properties of exponents to simplify the equation and isolate the variable ${ n }$. The final value of ${ n }$ is approximately 1.

Additional Information

It's worth noting that the equation ${ 9^n = 49 }$ has a limited range of solutions. Since ${ 9^n }$ is an increasing function, there is only one solution for ${ n }$ that satisfies the equation. Additionally, the solution is not an integer, which is a common property of exponential equations.

Final Answer

The final answer is: $\boxed{1}$

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Introduction

In our previous article, we solved for the variable ${ n }$ in the equation ${ 9^n = 49 }$. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q: What is the significance of the base 9 in the equation?

A: The base 9 in the equation is significant because it represents the exponential growth or decay of the variable ${ n }$. In this case, the base 9 is raised to the power of ${ n }$, which means that the value of ${ n }$ determines the rate at which the expression ${ 9^n }$ grows or decays.

Q: Why is the equation ${ 9^n = 49 }$ not straightforward to solve?

A: The equation ${ 9^n = 49 }$ is not straightforward to solve because it involves a non-integer base and a non-integer exponent. This makes it difficult to isolate the variable ${ n }$ using traditional algebraic methods.

Q: Can you explain the concept of logarithms in the context of this equation?

A: In the context of this equation, logarithms are used to solve for the variable ${ n }$. Specifically, we used the property of logarithms that states ${ a^x = b }$ if and only if ${ x = \log_a b }$. This property allows us to rewrite the equation in a form that is easier to solve.

Q: How do you calculate the value of ${ x }$ in the equation ${ x = \log_7 9 }$?

A: To calculate the value of ${ x }$, we can use the change of base formula, which states ${ \log_a b = \frac{\log_c b}{\log_c a} }$. Applying this formula to our equation, we get:

${ x = \frac{\log 9}{\log 7} }$

Q: What is the relationship between the value of ${ n }$ and the value of ${ x }$?

A: The value of ${ n }$ is approximately equal to the value of ${ x }$. In other words, ${ n \approx x }$. This is because we used the property of logarithms to solve for ${ x }$, and the value of ${ x }$ is approximately equal to the value of ${ n }$.

Q: Can you provide more information about the properties of exponents that were used in this equation?

A: Yes, certainly. The properties of exponents that were used in this equation include:

• The property that states ${ a^m = a^n }$ if and only if ${ m = n }$
• The property that states ${ a^m = b^n }$ if and only if ${ a = b^{m/n} }$
• The property that states ${ (a^m)^n = a^{mn} }$

These properties were used to simplify the equation and isolate the variable ${ n }$.

Q: What is the final answer to the equation ${ 9^n = 49 }$?

A: The final answer to the equation ${ 9^n = 49 }$ is approximately ${ n \approx 1 }$.

Q: Can you provide more information about the limitations of the solution?

A: Yes, certainly. The solution to the equation ${ 9^n = 49 }$ is limited to a single value of ${ n }$, which is approximately 1. This is because the equation is not an integer equation, and the solution is not an integer.

Q: What are some common applications of exponential equations like this one?

A: Exponential equations like this one have many common applications in fields such as finance, economics, and science. For example, they can be used to model population growth, chemical reactions, and financial investments.

Q: Can you provide more information about the concept of logarithms in general?

A: Yes, certainly. Logarithms are a mathematical concept that is used to solve equations involving exponents. They are defined as the inverse operation of exponentiation, and they have many important properties and applications.

Q: What are some common mistakes to avoid when solving exponential equations like this one?

A: Some common mistakes to avoid when solving exponential equations like this one include:

• Not using the correct properties of exponents
• Not simplifying the equation correctly
• Not isolating the variable correctly
• Not checking the solution for validity

By avoiding these common mistakes, you can ensure that your solution is accurate and reliable.