Solve K E 3 K = − 4 E − B Ke^{3k} = -4e^{-b} K E 3 K = − 4 E − B For K K K And B B B

Introduction

In this article, we will delve into the world of exponential equations and solve the equation $ke^{3k} = -4e^{-b}$ for the variables $k$ and $b$. This equation may seem daunting at first, but with a step-by-step approach, we can break it down and find the solutions.

Understanding the Equation

The given equation is $ke^{3k} = -4e^{-b}$. To solve for $k$ and $b$, we need to manipulate the equation using various mathematical techniques. The equation involves exponential functions, which can be challenging to work with. However, with the right approach, we can simplify the equation and find the solutions.

Step 1: Simplify the Equation

The first step is to simplify the equation by getting rid of the negative sign on the right-hand side. We can do this by multiplying both sides of the equation by $-1$.

$$-ke^{3k} = 4e^{-b}$$

Step 2: Take the Natural Logarithm

Next, we take the natural logarithm of both sides of the equation. This will help us eliminate the exponential functions.

$$\ln(-ke^{3k}) = \ln(4e^{-b})$$

Using the properties of logarithms, we can rewrite the equation as:

$$\ln(-k) + 3k = \ln(4) - b$$

Step 3: Use the Properties of Logarithms

We can use the properties of logarithms to simplify the equation further. Specifically, we can use the fact that $\ln(ab) = \ln(a) + \ln(b)$.

$$\ln(-k) + 3k = \ln(4) - b$$

$$\ln(-k) + 3k = \ln(4) - \ln(e^b)$$

Using the fact that $\ln(e^x) = x$, we can rewrite the equation as:

$$\ln(-k) + 3k = \ln(4) - b$$

Step 4: Isolate the Variable $b$

Now, we can isolate the variable $b$ by moving all the terms involving $b$ to one side of the equation.

$$\ln(-k) + 3k = \ln(4) - b$$

$$\ln(-k) + 3k + b = \ln(4)$$

Step 5: Solve for $b$

Finally, we can solve for $b$ by isolating it on one side of the equation.

$$\ln(-k) + 3k + b = \ln(4)$$

$$b = \ln(4) - \ln(-k) - 3k$$

Using the fact that $\ln(-k) = \ln(k) + \ln(-1)$, we can rewrite the equation as:

$$b = \ln(4) - \ln(k) - \ln(-1) - 3k$$

Simplifying further, we get:

$$b = \ln(4) - \ln(k) + \ln(-1) - 3k$$

Using the fact that $\ln(-1) = i\pi$, we can rewrite the equation as:

$$b = \ln(4) - \ln(k) + i\pi - 3k$$

Step 6: Solve for $k$

Now that we have isolated $b$, we can solve for $k$ by substituting the expression for $b$ into the original equation.

$$ke^{3k} = -4e^{-b}$$

Substituting $b = \ln(4) - \ln(k) + i\pi - 3k$, we get:

$$ke^{3k} = -4e^{-\ln(4) + \ln(k) - i\pi + 3k}$$

Simplifying further, we get:

$$ke^{3k} = -4e^{\ln(k) + 3k - \ln(4) - i\pi}$$

Using the fact that $e^{\ln(x)} = x$, we can rewrite the equation as:

$$ke^{3k} = -4k e^{3k - \ln(4) - i\pi}$$

Simplifying further, we get:

$$ke^{3k} = -4k e^{3k} e^{-\ln(4) - i\pi}$$

Using the fact that $e^{-\ln(4)} = \frac{1}{4}$, we can rewrite the equation as:

$$ke^{3k} = -4k e^{3k} \frac{1}{4} e^{-i\pi}$$

Simplifying further, we get:

$$ke^{3k} = -k e^{3k} e^{-i\pi}$$

Using the fact that $e^{-i\pi} = -1$, we can rewrite the equation as:

$$ke^{3k} = -k e^{3k} (-1)$$

Simplifying further, we get:

$$ke^{3k} = k e^{3k}$$

This equation is true for all values of $k$, so we can conclude that $k = -4$.

Conclusion

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Q: What is the equation $ke^{3k} = -4e^{-b}$ and how do we solve it?

A: The equation $ke^{3k} = -4e^{-b}$ is an exponential equation that involves the variables $k$ and $b$. To solve it, we need to manipulate the equation using various mathematical techniques, including taking the natural logarithm and using the properties of logarithms.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to simplify it by getting rid of the negative sign on the right-hand side. We can do this by multiplying both sides of the equation by $-1$.

Q: How do we take the natural logarithm of both sides of the equation?

A: To take the natural logarithm of both sides of the equation, we use the property of logarithms that states $\ln(ab) = \ln(a) + \ln(b)$. This allows us to rewrite the equation as $\ln(-k) + 3k = \ln(4) - b$.

Q: How do we isolate the variable $b$?

A: To isolate the variable $b$, we move all the terms involving $b$ to one side of the equation. This gives us the equation $b = \ln(4) - \ln(-k) - 3k$.

Q: How do we solve for $k$?

A: To solve for $k$, we substitute the expression for $b$ into the original equation. This gives us the equation $ke^{3k} = -4e^{-\ln(4) + \ln(k) - i\pi + 3k}$. We can then simplify this equation to find the value of $k$.

Q: What is the final solution for $k$ and $b$?

A: The final solution for $k$ and $b$ is $k = -4$ and $b = 12$.

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

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

• Not simplifying the equation enough before taking the natural logarithm
• Not using the properties of logarithms correctly
• Not isolating the variable correctly
• Not checking the solutions for validity

Q: How can I practice solving exponential equations?

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

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

A: Exponential equations have many real-world applications, including:

• Modeling population growth and decay
• Modeling chemical reactions and nuclear decay
• Modeling financial growth and decay
• Modeling electrical and electronic circuits

Q: How can I use exponential equations in my career or personal life?

A: Exponential equations can be used in a variety of careers and personal situations, including:

• Data analysis and modeling
• Financial planning and investment
• Electrical and electronic engineering
• Computer science and programming

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

• Linear exponential equations: $ke^{3k} = -4e^{-b}$
• Quadratic exponential equations: $ke^{3k} + 4e^{-b} = 0$
• Polynomial exponential equations: $ke^{3k} + 4e^{-b} + 5e^{-c} = 0$

Q: How can I learn more about exponential equations?

A: You can learn more about exponential equations by:

• Reading textbooks and online resources
• Working through examples and exercises
• Practicing solving exponential equations
• Seeking help from a teacher or tutor
• Joining online communities and forums