Which Of The Following Is A Solution To $11\left(4^{x+12}\right)-1=21$? (Rounded To Three Decimal Places).A. -11.5 B. -11.9 C. -10.9 D. -12.3

by ADMIN 147 views

Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore a solution to the equation $11\left(4^{x+12}\right)-1=21$ and provide a step-by-step guide on how to arrive at the answer.

Understanding Exponential Equations

Exponential equations involve variables in the exponent of a number. In this case, we have the equation $11\left(4^{x+12}\right)-1=21$. To solve this equation, we need to isolate the variable x.

Step 1: Isolate the Exponential Term

The first step is to isolate the exponential term on one side of the equation. We can do this by adding 1 to both sides of the equation:

11(4x+12)=2211\left(4^{x+12}\right) = 22

Step 2: Divide Both Sides by 11

Next, we need to divide both sides of the equation by 11 to isolate the exponential term:

4x+12=24^{x+12} = 2

Step 3: Take the Logarithm of Both Sides

To solve for x, we need to take the logarithm of both sides of the equation. We can use any base for the logarithm, but in this case, we will use the natural logarithm (ln):

ln⁡(4x+12)=ln⁡(2)\ln\left(4^{x+12}\right) = \ln(2)

Step 4: Use the Power Rule of Logarithms

The power rule of logarithms states that ln⁡(ab)=bln⁡(a)\ln(a^b) = b\ln(a). We can use this rule to simplify the left-hand side of the equation:

(x+12)ln⁡(4)=ln⁡(2)(x+12)\ln(4) = \ln(2)

Step 5: Divide Both Sides by ln(4)

Next, we need to divide both sides of the equation by ln(4) to solve for x+12:

x+12=ln⁡(2)ln⁡(4)x+12 = \frac{\ln(2)}{\ln(4)}

Step 6: Subtract 12 from Both Sides

Finally, we need to subtract 12 from both sides of the equation to solve for x:

x=ln⁡(2)ln⁡(4)−12x = \frac{\ln(2)}{\ln(4)} - 12

Calculating the Value of x

To calculate the value of x, we need to use a calculator to evaluate the expression ln⁡(2)ln⁡(4)−12\frac{\ln(2)}{\ln(4)} - 12. Plugging in the values, we get:

x≈0.6931.386−12x \approx \frac{0.693}{1.386} - 12

x≈−11.9x \approx -11.9

Conclusion

In this article, we have solved the exponential equation $11\left(4^{x+12}\right)-1=21$ and arrived at the solution x ≈ -11.9. This solution is rounded to three decimal places, as specified in the problem.

Answer

The correct answer is B. -11.9.

Discussion

This problem requires a good understanding of exponential equations and logarithms. The key to solving this problem is to isolate the exponential term and then use the power rule of logarithms to simplify the equation. With practice and patience, anyone can master the art of solving exponential equations.

Additional Resources

For more information on exponential equations and logarithms, check out the following resources:

  • Khan Academy: Exponential and Logarithmic Equations
  • Mathway: Exponential and Logarithmic Equations
  • Wolfram Alpha: Exponential and Logarithmic Equations

Final Thoughts

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent of a number. For example, the equation 2x=82^x = 8 is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term on one side of the equation. Then, you can use the power rule of logarithms to simplify the equation and solve for the variable.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that ln⁡(ab)=bln⁡(a)\ln(a^b) = b\ln(a). This means that you can simplify the logarithm of a power by multiplying the exponent by the logarithm of the base.

Q: How do I use the power rule of logarithms to solve an exponential equation?

A: To use the power rule of logarithms to solve an exponential equation, you need to take the logarithm of both sides of the equation. Then, you can use the power rule to simplify the equation and solve for the variable.

Q: What is the difference between an exponential equation and a logarithmic equation?

A: An exponential equation is an equation that involves a variable in the exponent of a number, while a logarithmic equation is an equation that involves a variable as the exponent of a number. For example, the equation 2x=82^x = 8 is an exponential equation, while the equation log⁡2(8)=x\log_2(8) = x is a logarithmic equation.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic term on one side of the equation. Then, you can use the definition of a logarithm to simplify the equation and solve for the variable.

Q: What is the definition of a logarithm?

A: The definition of a logarithm is that log⁡b(a)=c\log_b(a) = c is equivalent to bc=ab^c = a. This means that the logarithm of a number is the exponent to which the base must be raised to produce the number.

Q: How do I use the definition of a logarithm to solve a logarithmic equation?

A: To use the definition of a logarithm to solve a logarithmic equation, you need to rewrite the equation in exponential form. Then, you can solve for the variable.

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

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

  • Not isolating the exponential or logarithmic term on one side of the equation
  • Not using the power rule of logarithms correctly
  • Not using the definition of a logarithm correctly
  • Not checking the domain of the equation

Q: How can I practice solving exponential and logarithmic equations?

A: You can practice solving exponential and logarithmic equations by working through example problems and exercises. You can also use online resources, such as Khan Academy and Mathway, to practice solving exponential and logarithmic equations.

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

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

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

Conclusion

Exponential and logarithmic equations are an important part of mathematics, with many real-world applications. By understanding how to solve these types of equations, you can gain a deeper understanding of mathematical concepts and develop problem-solving skills that can be applied to a wide range of fields.