For The Equation $ac^* = D$, Solve For The Variable $t$ In Terms Of $ A , C A, C A , C [/tex], And $d$. Express Your Answer In Terms Of The Natural Logarithm.

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Introduction

In mathematics, equations involving variables and constants are a fundamental concept. Solving for a variable in an equation is a crucial skill that is used extensively in various fields, including physics, engineering, and economics. In this article, we will focus on solving for the variable t in the equation ac^* = d, where a, c, and d are constants. We will express our answer in terms of the natural logarithm.

Understanding the Equation

The given equation is ac^* = d. To solve for t, we need to isolate t on one side of the equation. However, the equation is not in a standard form, and we need to manipulate it to make it easier to solve.

Manipulating the Equation

To start solving the equation, we can take the natural logarithm of both sides. This will help us to eliminate the exponentiation and make the equation more manageable.

ln⁡(act)=ln⁡(d)\ln(ac^t) = \ln(d)

Using Logarithmic Properties

We can use the property of logarithms that states ln(ab) = ln(a) + ln(b). This will allow us to simplify the left-hand side of the equation.

ln⁡(a)+ln⁡(ct)=ln⁡(d)\ln(a) + \ln(c^t) = \ln(d)

Isolating the Variable t

Now, we can isolate the variable t by subtracting ln(a) from both sides of the equation.

ln⁡(ct)=ln⁡(d)−ln⁡(a)\ln(c^t) = \ln(d) - \ln(a)

Using Logarithmic Properties Again

We can use the property of logarithms that states ln(a^b) = b*ln(a). This will allow us to simplify the left-hand side of the equation.

tln⁡(c)=ln⁡(d)−ln⁡(a)t\ln(c) = \ln(d) - \ln(a)

Solving for t

Finally, we can solve for t by dividing both sides of the equation by ln(c).

t=ln⁡(d)−ln⁡(a)ln⁡(c)t = \frac{\ln(d) - \ln(a)}{\ln(c)}

Conclusion

In this article, we have solved for the variable t in the equation ac^* = d. We have expressed our answer in terms of the natural logarithm. The final solution is t = (ln(d) - ln(a))/ln(c). This equation can be used to solve for t in various mathematical and scientific applications.

Applications of the Equation

The equation ac^* = d has various applications in mathematics and science. For example, it can be used to model population growth, chemical reactions, and electrical circuits. The equation can also be used to solve problems involving exponential decay and growth.

Example Problems

Here are a few example problems that can be solved using the equation ac^* = d.

Example 1

Solve for t in the equation 2e^t = 5.

Solution

We can start by taking the natural logarithm of both sides.

ln⁡(2et)=ln⁡(5)\ln(2e^t) = \ln(5)

Using the property of logarithms, we can simplify the left-hand side.

ln⁡(2)+ln⁡(et)=ln⁡(5)\ln(2) + \ln(e^t) = \ln(5)

Subtracting ln(2) from both sides, we get:

ln⁡(et)=ln⁡(5)−ln⁡(2)\ln(e^t) = \ln(5) - \ln(2)

Using the property of logarithms again, we can simplify the left-hand side.

tln⁡(e)=ln⁡(5)−ln⁡(2)t\ln(e) = \ln(5) - \ln(2)

Since ln(e) = 1, we can simplify the equation further.

t=ln⁡(5)−ln⁡(2)t = \ln(5) - \ln(2)

Example 2

Solve for t in the equation 3e^2t = 10.

Solution

We can start by taking the natural logarithm of both sides.

ln⁡(3e2t)=ln⁡(10)\ln(3e^2t) = \ln(10)

Using the property of logarithms, we can simplify the left-hand side.

ln⁡(3)+ln⁡(e2t)=ln⁡(10)\ln(3) + \ln(e^2t) = \ln(10)

Subtracting ln(3) from both sides, we get:

ln⁡(e2t)=ln⁡(10)−ln⁡(3)\ln(e^2t) = \ln(10) - \ln(3)

Using the property of logarithms again, we can simplify the left-hand side.

2tln⁡(e)=ln⁡(10)−ln⁡(3)2t\ln(e) = \ln(10) - \ln(3)

Since ln(e) = 1, we can simplify the equation further.

2t=ln⁡(10)−ln⁡(3)2t = \ln(10) - \ln(3)

Dividing both sides by 2, we get:

t=ln⁡(10)−ln⁡(3)2t = \frac{\ln(10) - \ln(3)}{2}

Conclusion

In this article, we have solved for the variable t in the equation ac^* = d. We have expressed our answer in terms of the natural logarithm. The final solution is t = (ln(d) - ln(a))/ln(c). This equation can be used to solve for t in various mathematical and scientific applications. We have also provided example problems to demonstrate how to use the equation in practice.

Introduction

In our previous article, we solved for the variable t in the equation ac^* = d. We expressed our answer in terms of the natural logarithm. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the equation ac^* = d, and how is it used in mathematics?

A: The equation ac^* = d is a mathematical equation that involves variables and constants. It is used to model exponential growth and decay in various fields, including physics, engineering, and economics.

Q: How do I solve for t in the equation ac^* = d?

A: To solve for t, you need to take the natural logarithm of both sides of the equation. This will help you to eliminate the exponentiation and make the equation more manageable. You can then use logarithmic properties to simplify the equation and isolate the variable t.

Q: What is the final solution for t in the equation ac^* = d?

A: The final solution for t is t = (ln(d) - ln(a))/ln(c). This equation can be used to solve for t in various mathematical and scientific applications.

Q: Can you provide example problems to demonstrate how to use the equation in practice?

A: Yes, we have provided two example problems in our previous article. The first problem involves solving for t in the equation 2e^t = 5, and the second problem involves solving for t in the equation 3e^2t = 10.

Q: What are some common applications of the equation ac^* = d?

A: The equation ac^* = d has various applications in mathematics and science. It can be used to model population growth, chemical reactions, and electrical circuits. It can also be used to solve problems involving exponential decay and growth.

Q: Can I use the equation ac^* = d to solve problems involving logarithms?

A: Yes, the equation ac^* = d can be used to solve problems involving logarithms. By taking the natural logarithm of both sides of the equation, you can eliminate the exponentiation and make the equation more manageable.

Q: What is the difference between the equation ac^* = d and the equation a^t = d?

A: The equation ac^* = d involves an exponential function with a base of c, while the equation a^t = d involves an exponential function with a base of a. The two equations are related, but they have different forms and applications.

Q: Can I use the equation ac^* = d to solve problems involving finance?

A: Yes, the equation ac^* = d can be used to solve problems involving finance. For example, it can be used to model the growth of investments or the decay of assets over time.

Q: What are some common mistakes to avoid when solving for t in the equation ac^* = d?

A: Some common mistakes to avoid when solving for t include:

  • Not taking the natural logarithm of both sides of the equation
  • Not using logarithmic properties to simplify the equation
  • Not isolating the variable t
  • Not checking the units of the variables and constants

Conclusion

In this Q&A article, we have provided answers to common questions about solving for the variable t in the equation ac^* = d. We have also provided example problems and discussed common applications and mistakes to avoid. We hope that this article has helped to clarify any doubts or questions that readers may have.