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

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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 $11\left(4^{x+12}\right)-1=21$. We will break down the solution step by step, using mathematical concepts and formulas to arrive at the final 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=22114^{x+12} = \frac{22}{11}

Step 3: Simplify the Right-Hand Side

The right-hand side of the equation can be simplified by dividing 22 by 11:

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

Step 4: 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 let's use the natural logarithm (ln):

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

Step 5: 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 6: Divide Both Sides by ln(4)

Next, we need to divide both sides of the equation by ln(4) to isolate x:

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

Step 7: Simplify the Right-Hand Side

The right-hand side of the equation can be simplified by dividing ln(2) by ln(4):

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

Step 8: 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

Using a Calculator to Find the Value of x

To find the value of x, we can 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≈−11.9x \approx -11.9

Conclusion

In this article, we solved the exponential equation $11\left(4^{x+12}\right)-1=21$. We broke down the solution step by step, using mathematical concepts and formulas to arrive at the final answer. The value of x is approximately -11.9.

Answer

The correct answer is:

  • B. -11.9

Discussion

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 $11\left(4^{x+12}\right)-1=21$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable x. This can be done by using mathematical concepts and formulas, such as logarithms and the power rule of logarithms.

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 rule can be used to simplify the left-hand side of an exponential equation.

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 left-hand side of the equation.

Q: What is the natural logarithm (ln)?

A: The natural logarithm (ln) is a logarithm with a base of e. It is denoted by the symbol ln⁡\ln.

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

A: To use the natural logarithm to solve an exponential equation, you need to take the natural logarithm of both sides of the equation. Then, you can use the power rule of logarithms to simplify the left-hand side of the equation.

Q: What is the difference between the natural logarithm and the common logarithm?

A: The natural logarithm (ln) and the common logarithm (log) are both logarithms, but they have different bases. The natural logarithm has a base of e, while the common logarithm has a base of 10.

Q: How do I choose between the natural logarithm and the common logarithm?

A: You can choose between the natural logarithm and the common logarithm based on the problem you are trying to solve. If the problem involves the number e, you should use the natural logarithm. If the problem involves the number 10, you should use the common logarithm.

Q: What is the significance of the power rule of logarithms?

A: The power rule of logarithms is a fundamental concept in mathematics that allows us to simplify complex expressions involving logarithms. It is used extensively in calculus, algebra, and other branches of mathematics.

Q: How do I apply the power rule of logarithms in real-world problems?

A: The power rule of logarithms can be applied in a variety of real-world problems, such as finance, engineering, and science. For example, it can be used to model population growth, chemical reactions, and other phenomena that involve exponential growth or decay.

Q: What are some common applications of exponential equations?

A: Exponential equations have a wide range of applications in science, engineering, and finance. Some common applications include:

  • Modeling population growth and decay
  • Calculating compound interest
  • Analyzing chemical reactions
  • Modeling the spread of diseases
  • Predicting the behavior of complex systems

Q: How do I use exponential equations to model real-world problems?

A: To use exponential equations to model real-world problems, you need to identify the variables and parameters involved in the problem. Then, you can use mathematical concepts and formulas to develop a mathematical model of the problem.

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

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

  • Failing to isolate the variable x
  • Using the wrong logarithm (e.g. using the natural logarithm when the problem involves the number 10)
  • Failing to apply the power rule of logarithms correctly
  • Making errors when simplifying expressions involving logarithms

Q: How do I check my work when solving exponential equations?

A: To check your work when solving exponential equations, you need to plug your solution back into the original equation and verify that it is true. You can also use a calculator to check your work.