What Is The Solution To $\ln(x^2 - 16) = 0$?A. $x = \pm \sqrt{17}$ B. $x = \pm 4$ C. $x = \pm \sqrt{32}$

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What is the Solution to ln⁡(x2−16)=0\ln(x^2 - 16) = 0?

In this article, we will explore the solution to the equation ln⁡(x2−16)=0\ln(x^2 - 16) = 0. This equation involves a natural logarithm and a quadratic expression. To solve it, we need to understand the properties of logarithms and how to manipulate them to isolate the variable.

The given equation is ln⁡(x2−16)=0\ln(x^2 - 16) = 0. This equation states that the natural logarithm of the expression x2−16x^2 - 16 is equal to 0. To solve for xx, we need to find the value of xx that makes the expression inside the logarithm equal to 1, since ln⁡(1)=0\ln(1) = 0.

Before we proceed, let's recall some properties of logarithms. The natural logarithm function has the following properties:

  • ln⁡(1)=0\ln(1) = 0
  • ln⁡(a)=ln⁡(b)\ln(a) = \ln(b) if and only if a=ba = b
  • ln⁡(ab)=ln⁡(a)+ln⁡(b)\ln(ab) = \ln(a) + \ln(b)
  • ln⁡(a/b)=ln⁡(a)−ln⁡(b)\ln(a/b) = \ln(a) - \ln(b)

Now, let's solve the equation ln⁡(x2−16)=0\ln(x^2 - 16) = 0. Since ln⁡(1)=0\ln(1) = 0, we can set the expression inside the logarithm equal to 1:

x2−16=1x^2 - 16 = 1

To simplify the equation, we can add 16 to both sides:

x2=17x^2 = 17

Now, we can take the square root of both sides:

x=±17x = \pm \sqrt{17}

Therefore, the solution to the equation ln⁡(x2−16)=0\ln(x^2 - 16) = 0 is x=±17x = \pm \sqrt{17}. This is the only value of xx that makes the expression inside the logarithm equal to 1.

Let's compare our solution with the other options:

  • Option A: x=±17x = \pm \sqrt{17}
  • Option B: x=±4x = \pm 4
  • Option C: x=±32x = \pm \sqrt{32}

Our solution matches option A. Options B and C are not correct solutions to the equation.

The final answer is x=±17\boxed{x = \pm \sqrt{17}}.

Here are some additional tips and tricks to help you solve logarithmic equations:

  • Always check the domain of the logarithmic function to ensure that the expression inside the logarithm is positive.
  • Use the properties of logarithms to simplify the equation and isolate the variable.
  • Be careful when taking the square root of both sides, as this can introduce extraneous solutions.

By following these tips and tricks, you can become proficient in solving logarithmic equations and tackle more complex problems with confidence.
Q&A: Solving Logarithmic Equations

In our previous article, we explored the solution to the equation ln⁡(x2−16)=0\ln(x^2 - 16) = 0. We learned how to use the properties of logarithms to simplify the equation and isolate the variable. In this article, we will answer some frequently asked questions about solving logarithmic equations.

A: A logarithmic equation is an equation that involves a logarithmic function, such as ln⁡(x)=0\ln(x) = 0. An exponential equation, on the other hand, is an equation that involves an exponential function, such as 2x=82^x = 8. While both types of equations involve exponents, they are solved using different techniques.

A: To determine if a logarithmic equation is true or false, you need to check if the expression inside the logarithm is positive. If the expression is positive, then the equation is true. If the expression is negative, then the equation is false.

A: No, you cannot use the same techniques to solve logarithmic equations as you would to solve quadratic equations. Logarithmic equations require a different set of techniques, such as using the properties of logarithms to simplify the equation and isolate the variable.

A: A natural logarithm is a logarithm with base ee, where ee is a mathematical constant approximately equal to 2.718. A common logarithm, on the other hand, is a logarithm with base 10. While both types of logarithms are used in mathematics, the natural logarithm is more commonly used in calculus and other advanced math topics.

A: Yes, you can use a calculator to solve logarithmic equations. However, you need to be careful when using a calculator, as it may not always give you the correct answer. It's always a good idea to check your work by hand to ensure that the answer is correct.

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

  • Not checking the domain of the logarithmic function
  • Not using the properties of logarithms to simplify the equation
  • Not isolating the variable correctly
  • Not checking the answer by hand

Solving logarithmic equations can be challenging, but with practice and patience, you can become proficient in solving them. Remember to use the properties of logarithms to simplify the equation and isolate the variable, and always check your work by hand to ensure that the answer is correct.

Here are some additional tips and tricks to help you solve logarithmic equations:

  • Always check the domain of the logarithmic function to ensure that the expression inside the logarithm is positive.
  • Use the properties of logarithms to simplify the equation and isolate the variable.
  • Be careful when using a calculator, as it may not always give you the correct answer.
  • Check your work by hand to ensure that the answer is correct.

By following these tips and tricks, you can become proficient in solving logarithmic equations and tackle more complex problems with confidence.