What Is The Solution To The Equation Below? Round Your Answer To Two Decimal Places. 6 X = 66 6^x = 66 6 X = 66 A. X = 2.34 X = 2.34 X = 2.34 B. X = 0.43 X = 0.43 X = 0.43 C. X = 1.04 X = 1.04 X = 1.04 D. X = 2.40 X = 2.40 X = 2.40

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and logarithms. In this article, we will explore the solution to the equation $6^x = 66$ and provide a step-by-step guide on how to solve it.

What is an Exponential Equation?

An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. Exponential equations can be written in the form $a^x = b$, where $a$ and $b$ are constants.

The Equation $6^x = 66$

The equation $6^x = 66$ is a classic example of an exponential equation. To solve this equation, we need to find the value of $x$ that satisfies the equation.

Step 1: Take the Logarithm of Both Sides

One way to solve exponential equations is to take the logarithm of both sides. This will allow us to use the properties of logarithms to simplify the equation.

$$\log(6^x) = \log(66)$$

Using the property of logarithms that states $\log(a^b) = b\log(a)$, we can rewrite the equation as:

$$x\log(6) = \log(66)$$

Step 2: Simplify the Equation

Now that we have the equation in the form $x\log(6) = \log(66)$, we can simplify it by dividing both sides by $\log(6)$.

$$x = \frac{\log(66)}{\log(6)}$$

Step 3: Evaluate the Expression

To find the value of $x$, we need to evaluate the expression $\frac{\log(66)}{\log(6)}$. We can use a calculator to find the values of $\log(66)$ and $\log(6)$.

$$\log(66) \approx 1.8235$$

$$\log(6) \approx 0.7782$$

Now, we can substitute these values into the expression:

$$x \approx \frac{1.8235}{0.7782}$$

Step 4: Round the Answer to Two Decimal Places

Finally, we need to round the answer to two decimal places. Using a calculator, we get:

$$x \approx 2.34$$

Conclusion

In this article, we solved the equation $6^x = 66$ using the properties of logarithms. We took the logarithm of both sides, simplified the equation, and evaluated the expression to find the value of $x$. The final answer is $x \approx 2.34$.

Comparison of Options

Now that we have the solution to the equation, let's compare it to the options provided:

• A. $x = 2.34$
• B. $x = 0.43$
• C. $x = 1.04$
• D. $x = 2.40$

Our solution, $x \approx 2.34$, matches option A.

Final Answer

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Introduction

Exponential equations can be a challenging topic for many students. In this article, we will answer some of the most frequently asked questions about exponential equations, including how to solve them, what they are, and how to use logarithms to simplify them.

Q: What is an Exponential Equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.

Q: How Do I Solve an Exponential Equation?

A: To solve an exponential equation, you need to isolate the variable $x$. One way to do this is to take the logarithm of both sides of the equation. This will allow you to use the properties of logarithms to simplify the equation.

Q: What is the Logarithm of an Exponential Function?

A: The logarithm of an exponential function is the inverse of the exponential function. In other words, if $f(x) = a^x$, then $\log(f(x)) = x$. This means that if you take the logarithm of both sides of an exponential equation, you can simplify the equation and isolate the variable $x$.

Q: How Do I Use Logarithms to Simplify an Exponential Equation?

A: To use logarithms to simplify an exponential equation, you need to take the logarithm of both sides of the equation. This will allow you to use the properties of logarithms to simplify the equation. For example, if you have the equation $a^x = b$, you can take the logarithm of both sides and get:

$$\log(a^x) = \log(b)$$

Using the property of logarithms that states $\log(a^b) = b\log(a)$, you can rewrite the equation as:

$$x\log(a) = \log(b)$$

Now, you can divide both sides of the equation by $\log(a)$ to get:

$$x = \frac{\log(b)}{\log(a)}$$

Q: What is the Difference Between a Logarithmic and Exponential Function?

A: A logarithmic function is the inverse of an exponential function. In other words, if $f(x) = a^x$, then $\log(f(x)) = x$. This means that if you take the logarithm of both sides of an exponential equation, you can simplify the equation and isolate the variable $x$.

Q: How Do I Use a Calculator to Solve an Exponential Equation?

A: To use a calculator to solve an exponential equation, you need to enter the equation into the calculator and use the logarithm function to simplify the equation. For example, if you have the equation $a^x = b$, you can enter the equation into the calculator and get:

$$x = \log_b(a)$$

Using the calculator, you can find the value of $x$ by entering the values of $a$ and $b$ into the calculator.

Q: What is the Final Answer to the Equation $6^x = 66$?

A: The final answer to the equation $6^x = 66$ is $x \approx 2.34$.

Conclusion

In this article, we answered some of the most frequently asked questions about exponential equations, including how to solve them, what they are, and how to use logarithms to simplify them. We also provided a step-by-step guide on how to solve the equation $6^x = 66$ using logarithms.