Solve The Equation 7 Log ⁡ 6 ( 9 X ) = 14 7 \log _6(9x) = 14 7 Lo G 6 ​ ( 9 X ) = 14 For X X X . X = X = \quad X = Enter Your Answer.

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Introduction

In this article, we will delve into the world of logarithms and explore how to solve an equation involving logarithms. The given equation is $7 \log _6(9x) = 14$, and our goal is to find the value of $x$ that satisfies this equation. We will use the properties of logarithms to simplify the equation and isolate the variable $x$.

Understanding Logarithms

Before we dive into solving the equation, let's take a moment to understand what logarithms are. A logarithm is the inverse operation of exponentiation. In other words, if we have a number $y$ that is the result of raising a base $b$ to a power $x$, then the logarithm of $y$ with base $b$ is equal to $x$. This can be expressed mathematically as:

$$b^x = y \Rightarrow x = \log_b(y)$$

For example, if we have $2^3 = 8$, then the logarithm of $8$ with base $2$ is equal to $3$, which can be written as:

$$\log_2(8) = 3$$

Simplifying the Equation

Now that we have a basic understanding of logarithms, let's simplify the given equation. We can start by using the property of logarithms that states:

$$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$$

where $c$ is any positive real number. We can apply this property to the given equation by rewriting it as:

$$7 \log _6(9x) = 14 \Rightarrow \log _6(9x) = \frac{14}{7} = 2$$

Isolating the Variable $x$

Now that we have simplified the equation, let's isolate the variable $x$. We can start by rewriting the equation in exponential form:

$$\log _6(9x) = 2 \Rightarrow 6^2 = 9x$$

Solving for $x$

Now that we have isolated the variable $x$, let's solve for its value. We can start by dividing both sides of the equation by $9$:

$$\frac{6^2}{9} = x \Rightarrow \frac{36}{9} = x$$

Conclusion

In this article, we have solved the equation $7 \log _6(9x) = 14$ for $x$. We started by simplifying the equation using the properties of logarithms, and then isolated the variable $x$ by rewriting the equation in exponential form. Finally, we solved for the value of $x$ by dividing both sides of the equation by $9$. The final answer is:

$x = \boxed{\frac{36}{9}}$

$x = \boxed{4}$

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Introduction

In our previous article, we solved the equation $7 \log _6(9x) = 14$ for $x$. We used the properties of logarithms to simplify the equation and isolate the variable $x$. In this article, we will answer some of the most frequently asked questions about solving this equation.

Q: What is the definition of a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if we have a number $y$ that is the result of raising a base $b$ to a power $x$, then the logarithm of $y$ with base $b$ is equal to $x$. This can be expressed mathematically as:

$$b^x = y \Rightarrow x = \log_b(y)$$

Q: How do I simplify the equation $7 \log _6(9x) = 14$?

A: To simplify the equation, we can use the property of logarithms that states:

$$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$$

where $c$ is any positive real number. We can apply this property to the given equation by rewriting it as:

$$7 \log _6(9x) = 14 \Rightarrow \log _6(9x) = \frac{14}{7} = 2$$

Q: How do I isolate the variable $x$ in the equation?

A: To isolate the variable $x$, we can rewrite the equation in exponential form:

$$\log _6(9x) = 2 \Rightarrow 6^2 = 9x$$

Q: How do I solve for $x$ in the equation?

A: To solve for $x$, we can divide both sides of the equation by $9$:

$$\frac{6^2}{9} = x \Rightarrow \frac{36}{9} = x$$

Q: What is the final answer to the equation?

A: The final answer to the equation is:

$x = \boxed{\frac{36}{9}}$

$x = \boxed{4}$

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

A: Yes, you can use a calculator to solve the equation. However, it's always a good idea to understand the underlying math and how to solve the equation by hand.

Q: What if I have a different equation to solve?

A: If you have a different equation to solve, you can try using the same techniques and properties of logarithms that we used in this article. If you're still having trouble, you can try searching online for help or seeking assistance from a math tutor or teacher.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving the equation $7 \log _6(9x) = 14$ for $x$. We have provided step-by-step instructions and explanations for each question, and have also provided the final answer to the equation. We hope that this article has been helpful in understanding how to solve this equation and other similar equations.

Additional Resources

If you're interested in learning more about logarithms and how to solve equations involving logarithms, we recommend checking out the following resources:

• Khan Academy: Logarithms
• Mathway: Logarithm Calculator
• Wolfram Alpha: Logarithm Calculator

We hope that these resources are helpful in your math journey!