Solve The Exponential Equation Using The Natural Logarithm.$\[ 8^x = 9 \\]What Is The Exact Answer? Select The Correct Choice Below And, If Necessary, Fill In The Answer.

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 the natural logarithm. We will use the equation $8^x = 9$ as an example to demonstrate the steps involved in solving exponential equations.

What is the Natural Logarithm?

The natural logarithm, denoted by $\ln(x)$, is the inverse function of the exponential function $e^x$. It is a fundamental concept in mathematics and is used to solve exponential equations. The natural logarithm has several properties that make it useful in solving exponential equations.

Properties of the Natural Logarithm

The natural logarithm has several properties that make it useful in solving exponential equations. Some of the key properties are:

• One-to-one correspondence: The natural logarithm is a one-to-one function, which means that it passes the horizontal line test. This means that for every unique input, there is a unique output.
• Inverse function: The natural logarithm is the inverse function of the exponential function $e^x$. This means that $\ln(e^x) = x$ and $e^{\ln(x)} = x$.
• Domain and range: The domain of the natural logarithm is all positive real numbers, and the range is all real numbers.

Solving Exponential Equations Using the Natural Logarithm

To solve exponential equations using the natural logarithm, we need to follow these steps:

1. Take the natural logarithm of both sides: We take the natural logarithm of both sides of the equation to get rid of the exponential term.
2. Use the properties of the natural logarithm: We use the properties of the natural logarithm to simplify the equation and isolate the variable.
3. Solve for the variable: We solve for the variable by isolating it on one side of the equation.

Step 1: Take the Natural Logarithm of Both Sides

To solve the equation $8^x = 9$, we take the natural logarithm of both sides:

$$\ln(8^x) = \ln(9)$$

Using the property of the natural logarithm that $\ln(a^b) = b\ln(a)$, we can simplify the equation:

$$x\ln(8) = \ln(9)$$

Step 2: Use the Properties of the Natural Logarithm

We can use the properties of the natural logarithm to simplify the equation further:

$$x\ln(8) = \ln(9)$$

$$x = \frac{\ln(9)}{\ln(8)}$$

Step 3: Solve for the Variable

We can solve for the variable by isolating it on one side of the equation:

$$x = \frac{\ln(9)}{\ln(8)}$$

Conclusion

In this article, we have demonstrated how to solve exponential equations using the natural logarithm. We have used the equation $8^x = 9$ as an example to demonstrate the steps involved in solving exponential equations. We have taken the natural logarithm of both sides, used the properties of the natural logarithm to simplify the equation, and solved for the variable.

Final Answer

The final answer is $\boxed{\frac{\ln(9)}{\ln(8)}}$.

Discussion

This problem is a classic example of how to solve exponential equations using the natural logarithm. The natural logarithm is a powerful tool that can be used to solve exponential equations, and it is an essential concept in mathematics.

Related Problems

• Solve the exponential equation $2^x = 5$ using the natural logarithm.
• Solve the exponential equation $3^x = 7$ using the natural logarithm.
• Solve the exponential equation $4^x = 11$ using the natural logarithm.

References

• "Exponential Functions" by Math Open Reference
• "Natural Logarithm" by Wolfram MathWorld
• "Solving Exponential Equations" by Khan Academy

Keywords

• Exponential equations
• Natural logarithm
• Inverse function
• Domain and range
• Solving exponential equations
  Solving Exponential Equations Using the Natural Logarithm: Q&A ===========================================================

Introduction

In our previous article, we demonstrated how to solve exponential equations using the natural logarithm. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on solving exponential equations using the natural logarithm.

Q: What is the natural logarithm?

A: The natural logarithm, denoted by $\ln(x)$, is the inverse function of the exponential function $e^x$. It is a fundamental concept in mathematics and is used to solve exponential equations.

Q: What are the properties of the natural logarithm?

A: The natural logarithm has several properties that make it useful in solving exponential equations. Some of the key properties are:

• One-to-one correspondence: The natural logarithm is a one-to-one function, which means that it passes the horizontal line test. This means that for every unique input, there is a unique output.
• Inverse function: The natural logarithm is the inverse function of the exponential function $e^x$. This means that $\ln(e^x) = x$ and $e^{\ln(x)} = x$.
• Domain and range: The domain of the natural logarithm is all positive real numbers, and the range is all real numbers.

Q: How do I solve exponential equations using the natural logarithm?

A: To solve exponential equations using the natural logarithm, you need to follow these steps:

1. Take the natural logarithm of both sides: We take the natural logarithm of both sides of the equation to get rid of the exponential term.
2. Use the properties of the natural logarithm: We use the properties of the natural logarithm to simplify the equation and isolate the variable.
3. Solve for the variable: We solve for the variable by isolating it on one side of the equation.

Q: What if the equation has a base that is not a power of e?

A: If the equation has a base that is not a power of e, we can use the change of base formula to rewrite the equation in terms of the natural logarithm. The change of base formula is:

$$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$$

where $c$ is any positive real number.

Q: Can I use the natural logarithm to solve exponential equations with negative bases?

A: Yes, you can use the natural logarithm to solve exponential equations with negative bases. However, you need to be careful when taking the natural logarithm of a negative number, as it will result in a complex number.

Q: What are some common mistakes to avoid when solving exponential equations using the natural logarithm?

A: Some common mistakes to avoid when solving exponential equations using the natural logarithm are:

• Not taking the natural logarithm of both sides: Make sure to take the natural logarithm of both sides of the equation to get rid of the exponential term.
• Not using the properties of the natural logarithm: Make sure to use the properties of the natural logarithm to simplify the equation and isolate the variable.
• Not solving for the variable: Make sure to solve for the variable by isolating it on one side of the equation.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts and provide additional information on solving exponential equations using the natural logarithm. We have covered topics such as the properties of the natural logarithm, how to solve exponential equations using the natural logarithm, and common mistakes to avoid.

Final Answer

The final answer is $\boxed{\frac{\ln(9)}{\ln(8)}}$.

Discussion

This problem is a classic example of how to solve exponential equations using the natural logarithm. The natural logarithm is a powerful tool that can be used to solve exponential equations, and it is an essential concept in mathematics.

Related Problems

• Solve the exponential equation $2^x = 5$ using the natural logarithm.
• Solve the exponential equation $3^x = 7$ using the natural logarithm.
• Solve the exponential equation $4^x = 11$ using the natural logarithm.

References

• "Exponential Functions" by Math Open Reference
• "Natural Logarithm" by Wolfram MathWorld
• "Solving Exponential Equations" by Khan Academy

Keywords

• Exponential equations
• Natural logarithm
• Inverse function
• Domain and range
• Solving exponential equations