Solve 6 ( 2 ) X = 384 6(2)^x = 384 6 ( 2 ) X = 384 .

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Introduction

Solving exponential equations can be a challenging task, especially when dealing with variables in the exponent. In this article, we will focus on solving the equation $6(2)^x = 384$ using various methods and techniques. We will start by understanding the properties of exponential functions and then apply these properties to solve the given equation.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes a relationship between two variables, where one variable is the exponent of the other. The general form of an exponential function is $f(x) = a^x$, where $a$ is the base and $x$ is the exponent. In the given equation, the base is $2$ and the exponent is $x$. The equation can be rewritten as $2^x = \frac{384}{6}$.

Simplifying the Equation

To simplify the equation, we can start by dividing both sides by $6$. This gives us $2^x = 64$. Now, we can see that the base is $2$ and the result is $64$, which is a power of $2$. Specifically, $64 = 2^6$. Therefore, we can rewrite the equation as $2^x = 2^6$.

Equating Exponents

Since the bases are the same, we can equate the exponents. This gives us $x = 6$. Therefore, the solution to the equation $6(2)^x = 384$ is $x = 6$.

Verifying the Solution

To verify the solution, we can substitute $x = 6$ back into the original equation. This gives us $6(2)^6 = 6 \cdot 64 = 384$. Therefore, the solution $x = 6$ satisfies the original equation.

Alternative Methods

There are alternative methods to solve the equation $6(2)^x = 384$. One method is to use logarithms. We can take the logarithm of both sides of the equation, which gives us $\log(6(2)^x) = \log(384)$. Using the property of logarithms, we can rewrite this as $\log(6) + x \log(2) = \log(384)$. Now, we can solve for $x$ by isolating the term with the variable.

Using Logarithms

To use logarithms, we can start by taking the logarithm of both sides of the equation. This gives us $\log(6(2)^x) = \log(384)$. Using the property of logarithms, we can rewrite this as $\log(6) + x \log(2) = \log(384)$. Now, we can solve for $x$ by isolating the term with the variable.

Isolating the Variable

To isolate the variable, we can start by subtracting $\log(6)$ from both sides of the equation. This gives us $x \log(2) = \log(384) - \log(6)$. Now, we can divide both sides by $\log(2)$ to solve for $x$.

Solving for x

To solve for $x$, we can start by dividing both sides of the equation by $\log(2)$. This gives us $x = \frac{\log(384) - \log(6)}{\log(2)}$. Now, we can simplify the expression by using the properties of logarithms.

Simplifying the Expression

To simplify the expression, we can start by using the property of logarithms that states $\log(a) - \log(b) = \log(\frac{a}{b})$. This gives us $x = \frac{\log(\frac{384}{6})}{\log(2)}$. Now, we can simplify the expression by evaluating the logarithm.

Evaluating the Logarithm

To evaluate the logarithm, we can start by simplifying the fraction inside the logarithm. This gives us $x = \frac{\log(64)}{\log(2)}$. Now, we can use the property of logarithms that states $\log(a^b) = b \log(a)$.

Using the Property of Logarithms

To use the property of logarithms, we can start by rewriting the logarithm as an exponent. This gives us $x = \frac{6 \log(2)}{\log(2)}$. Now, we can simplify the expression by canceling out the common factor.

Canceling Out the Common Factor

To cancel out the common factor, we can start by dividing both sides of the equation by $\log(2)$. This gives us $x = 6$. Therefore, the solution to the equation $6(2)^x = 384$ is $x = 6$.

Conclusion

In this article, we have solved the equation $6(2)^x = 384$ using various methods and techniques. We have used the properties of exponential functions, logarithms, and algebraic manipulations to isolate the variable and solve for $x$. The solution to the equation is $x = 6$, which satisfies the original equation.

Final Answer

The final answer to the equation $6(2)^x = 384$ is $x = 6$.

Introduction

In our previous article, we solved the equation $6(2)^x = 384$ using various methods and techniques. In this article, we will answer some frequently asked questions related to solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is a type of mathematical equation that involves an exponential function. The general form of an exponential equation is $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use various methods and techniques, such as:

• Using the properties of exponential functions
• Taking the logarithm of both sides of the equation
• Using algebraic manipulations to isolate the variable
• Using a calculator to evaluate the exponential function

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

A: An exponential equation involves an exponential function, whereas a linear equation involves a linear function. Exponential equations can be more challenging to solve than linear equations, especially when dealing with variables in the exponent.

Q: Can I use a calculator to solve an exponential equation?

A: Yes, you can use a calculator to evaluate the exponential function and solve the equation. However, it's always a good idea to understand the underlying mathematics and use a calculator as a tool to verify your solution.

Q: How do I know if an equation is exponential or linear?

A: To determine if an equation is exponential or linear, look for the presence of an exponential function. If the equation involves a base raised to a power, it's likely an exponential equation. If the equation involves a linear function, it's likely a linear equation.

Q: Can I use logarithms to solve an exponential equation?

A: Yes, you can use logarithms to solve an exponential equation. Taking the logarithm of both sides of the equation can help you isolate the variable and solve for the exponent.

Q: What is the property of logarithms that states $\log(a) - \log(b) = \log(\frac{a}{b})$?

A: This property of logarithms states that the difference between two logarithms is equal to the logarithm of the ratio of the two numbers. This property can be useful when simplifying expressions involving logarithms.

Q: How do I use the property of logarithms to simplify an expression?

A: To use the property of logarithms to simplify an expression, look for opportunities to rewrite the expression as a ratio of two numbers. Then, apply the property of logarithms to simplify the expression.

Q: Can I use a graphing calculator to solve an exponential equation?

A: Yes, you can use a graphing calculator to solve an exponential equation. Graphing calculators can help you visualize the relationship between the variables and find the solution to the equation.

Q: How do I know if my solution to an exponential equation is correct?

A: To verify your solution to an exponential equation, substitute the solution back into the original equation and check if it's true. If the solution satisfies the original equation, it's likely correct.

Conclusion

In this article, we have answered some frequently asked questions related to solving exponential equations. We have covered topics such as the definition of an exponential equation, methods for solving exponential equations, and properties of logarithms. We hope this article has been helpful in clarifying any doubts you may have had about solving exponential equations.