What Is The Solution To The Equation Below? Round Your Answer To Two Decimal Places. 4 ⋅ E X = 12.76 4 \cdot E^x = 12.76 4 ⋅ E X = 12.76 A. X = 1.16 X = 1.16 X = 1.16 B. X = 23.40 X = 23.40 X = 23.40 C. X = 0.50 X = 0.50 X = 0.50 D. X = 24.29 X = 24.29 X = 24.29

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Introduction

In this article, we will explore the solution to the equation $4 \cdot e^x = 12.76$. This equation involves an exponential function, which is a fundamental concept in mathematics. The exponential function is a function that grows or decays exponentially, and it is a crucial tool in many areas of mathematics, science, and engineering.

Understanding the Equation

The equation $4 \cdot e^x = 12.76$ involves the exponential function $e^x$, which is a mathematical constant approximately equal to $2.71828$. The equation states that the product of $4$ and the exponential function $e^x$ is equal to $12.76$. To solve this equation, we need to isolate the variable $x$.

Isolating the Variable $x$

To isolate the variable $x$, we can start by dividing both sides of the equation by $4$. This will give us:

$$e^x = \frac{12.76}{4}$$

Simplifying the Equation

Now, we can simplify the equation by evaluating the right-hand side:

$$e^x = 3.19$$

Using the Natural Logarithm

To solve for $x$, we can use the natural logarithm (ln) function, which is the inverse of the exponential function. The natural logarithm of a number is the power to which the base $e$ must be raised to produce that number. In this case, we can take the natural logarithm of both sides of the equation:

$$\ln(e^x) = \ln(3.19)$$

Simplifying the Equation

Using the property of logarithms that states $\ln(e^x) = x$, we can simplify the equation:

$$x = \ln(3.19)$$

Evaluating the Natural Logarithm

Now, we can evaluate the natural logarithm of $3.19$:

$$x = 1.16$$

Conclusion

In conclusion, the solution to the equation $4 \cdot e^x = 12.76$ is $x = 1.16$. This solution is obtained by isolating the variable $x$, using the natural logarithm function, and evaluating the natural logarithm of $3.19$.

Comparison with Other Options

Let's compare our solution with the other options:

• Option A: $x = 1.16$
• Option B: $x = 23.40$
• Option C: $x = 0.50$
• Option D: $x = 24.29$

Our solution, $x = 1.16$, is the correct answer. The other options are incorrect.

Importance of the Exponential Function

The exponential function is a fundamental concept in mathematics, and it has many important applications in science and engineering. The exponential function is used to model population growth, chemical reactions, and electrical circuits, among other things. In this article, we have seen how the exponential function can be used to solve equations involving exponential functions.

Real-World Applications

The exponential function has many real-world applications, including:

• Modeling population growth: The exponential function can be used to model the growth of populations, such as the growth of bacteria or the growth of a city.
• Chemical reactions: The exponential function can be used to model chemical reactions, such as the decay of radioactive materials.
• Electrical circuits: The exponential function can be used to model the behavior of electrical circuits, such as the decay of electrical signals.

Conclusion

In conclusion, the solution to the equation $4 \cdot e^x = 12.76$ is $x = 1.16$. This solution is obtained by isolating the variable $x$, using the natural logarithm function, and evaluating the natural logarithm of $3.19$. The exponential function is a fundamental concept in mathematics, and it has many important applications in science and engineering.

Final Answer

The final answer is $x = 1.16$.

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Introduction

In this article, we will answer some frequently asked questions (FAQs) about the equation $4 \cdot e^x = 12.76$. This equation involves an exponential function, which is a fundamental concept in mathematics. The exponential function is a function that grows or decays exponentially, and it is a crucial tool in many areas of mathematics, science, and engineering.

Q: What is the exponential function?

A: The exponential function is a mathematical function that grows or decays exponentially. It is a function that is defined as $e^x$, where $e$ is a mathematical constant approximately equal to $2.71828$.

Q: What is the natural logarithm function?

A: The natural logarithm function is the inverse of the exponential function. It is a function that is defined as $\ln(x)$, where $x$ is a positive real number.

Q: How do I solve the equation $4 \cdot e^x = 12.76$?

A: To solve the equation $4 \cdot e^x = 12.76$, you can start by dividing both sides of the equation by $4$. This will give you $e^x = 3.19$. Then, you can take the natural logarithm of both sides of the equation to get $x = \ln(3.19)$.

Q: What is the value of $x$ in the equation $4 \cdot e^x = 12.76$?

A: The value of $x$ in the equation $4 \cdot e^x = 12.76$ is $x = 1.16$.

Q: Why do we use the natural logarithm function to solve the equation $4 \cdot e^x = 12.76$?

A: We use the natural logarithm function to solve the equation $4 \cdot e^x = 12.76$ because it is the inverse of the exponential function. This means that if we take the natural logarithm of both sides of the equation, we can isolate the variable $x$.

Q: What are some real-world applications of the exponential function?

A: The exponential function has many real-world applications, including:

• Modeling population growth: The exponential function can be used to model the growth of populations, such as the growth of bacteria or the growth of a city.
• Chemical reactions: The exponential function can be used to model chemical reactions, such as the decay of radioactive materials.
• Electrical circuits: The exponential function can be used to model the behavior of electrical circuits, such as the decay of electrical signals.

Q: How do I evaluate the natural logarithm of a number?

A: To evaluate the natural logarithm of a number, you can use a calculator or a computer program. Alternatively, you can use a table of natural logarithms or a mathematical formula to approximate the value of the natural logarithm.

Q: What is the difference between the natural logarithm function and the logarithm function with base 10?

A: The natural logarithm function and the logarithm function with base 10 are both logarithmic functions, but they have different bases. The natural logarithm function has a base of $e$, while the logarithm function with base 10 has a base of 10.

Q: How do I convert between the natural logarithm function and the logarithm function with base 10?

A: To convert between the natural logarithm function and the logarithm function with base 10, you can use the following formula:

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

Conclusion

In conclusion, we have answered some frequently asked questions (FAQs) about the equation $4 \cdot e^x = 12.76$. This equation involves an exponential function, which is a fundamental concept in mathematics. The exponential function is a function that grows or decays exponentially, and it is a crucial tool in many areas of mathematics, science, and engineering.

Final Answer

The final answer is $x = 1.16$.