Use Natural Logarithms To Solve The Equation. Round To The Nearest Thousandth.${ 6e^{4x} - 2 = 3 }$A. -0.448 B. -0.046 C. 0.327 D. 0.067

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them is crucial in various fields such as science, engineering, and economics. In this article, we will focus on solving exponential equations using natural logarithms. We will use the given equation ${ 6e^{4x} - 2 = 3 }$ as an example to demonstrate the steps involved in solving exponential equations with natural logarithms.

Understanding Exponential Equations

Exponential equations are equations that involve 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. In the given equation, the exponential function is ${ e^{4x} }$, where ${ e }$ is the base of the natural logarithm.

Step 1: Isolate the Exponential Term

The first step in solving the equation is to isolate the exponential term. To do this, we add 2 to both sides of the equation:

${ 6e^{4x} - 2 + 2 = 3 + 2 }$

This simplifies to:

${ 6e^{4x} = 5 }$

Step 2: Divide Both Sides by 6

Next, we divide both sides of the equation by 6 to isolate the exponential term:

${ \frac{6e^{4x}}{6} = \frac{5}{6} }$

This simplifies to:

${ e^{4x} = \frac{5}{6} }$

Step 3: Take the Natural Logarithm of Both Sides

Now, we take the natural logarithm of both sides of the equation to eliminate the exponential term:

${ \ln(e^{4x}) = \ln\left(\frac{5}{6}\right) }$

Using the property of logarithms that states ${ \ln(a^b) = b\ln(a) }$, we can simplify the left-hand side of the equation:

${ 4x\ln(e) = \ln\left(\frac{5}{6}\right) }$

Since ${ \ln(e) = 1 }$, we can simplify the equation further:

${ 4x = \ln\left(\frac{5}{6}\right) }$

Step 4: Divide Both Sides by 4

Finally, we divide both sides of the equation by 4 to solve for ${ x }$:

${ \frac{4x}{4} = \frac{\ln\left(\frac{5}{6}\right)}{4} }$

This simplifies to:

${ x = \frac{\ln\left(\frac{5}{6}\right)}{4} }$

Rounding to the Nearest Thousandth

To round the answer to the nearest thousandth, we use a calculator to evaluate the expression:

${ x \approx \frac{\ln\left(\frac{5}{6}\right)}{4} }$

Using a calculator, we get:

${ x \approx -0.046 }$

Conclusion

In this article, we used natural logarithms to solve the equation ${ 6e^{4x} - 2 = 3 }$. We isolated the exponential term, divided both sides by 6, took the natural logarithm of both sides, and finally divided both sides by 4 to solve for ${ x }$. The final answer is ${ x \approx -0.046 }$, which is the correct answer among the given options.

Answer Key

The correct answer is:

• B. -0.046

Discussion

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: What is a natural logarithm?

A: A natural logarithm is the logarithm of a number to the base ${ e }$, where ${ e }$ is a mathematical constant approximately equal to 2.71828.

Q: How do I isolate the exponential term in an exponential equation?

A: To isolate the exponential term, you need to add or subtract the same value from both sides of the equation. For example, if the equation is ${ 6e^{4x} - 2 = 3 }$, you can add 2 to both sides to get ${ 6e^{4x} = 5 }$.

Q: How do I use natural logarithms to solve an exponential equation?

A: To use natural logarithms to solve an exponential equation, you need to take the natural logarithm of both sides of the equation. This will eliminate the exponential term and allow you to solve for the variable.

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

A: This property states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number. For example, ${ \ln(e^x) = x\ln(e) }$.

Q: How do I use a calculator to evaluate an expression involving a natural logarithm?

A: To use a calculator to evaluate an expression involving a natural logarithm, you need to enter the expression into the calculator and press the "log" or "ln" button. For example, to evaluate the expression ${ \ln\left(\frac{5}{6}\right) }$, you would enter the expression into the calculator and press the "log" or "ln" button.

Q: What is the final answer to the equation ${ 6e^{4x} - 2 = 3 }$?

A: The final answer to the equation ${ 6e^{4x} - 2 = 3 }$ is ${ x \approx -0.046 }$.

Q: What are some common mistakes to avoid when solving exponential equations with natural logarithms?

A: Some common mistakes to avoid when solving exponential equations with natural logarithms include:

• Not isolating the exponential term
• Not taking the natural logarithm of both sides of the equation
• Not using the correct property of logarithms
• Not using a calculator to evaluate expressions involving natural logarithms

Q: How can I practice solving exponential equations with natural logarithms?

A: You can practice solving exponential equations with natural logarithms by working through examples and exercises in a textbook or online resource. You can also try solving real-world problems that involve exponential equations with natural logarithms.

Conclusion

Solving exponential equations with natural logarithms is a powerful tool for solving equations that involve exponential functions. By following the steps outlined in this article, you can solve exponential equations with natural logarithms and gain a deeper understanding of the underlying mathematics.