Converting Logarithmic to Exponential Form

Oct 13, 2025 by miodeya.com 19 views

Understanding the Problem

The problem asks us to find an equivalent equation to the given logarithmic equation $\log _x 36 = 2$ . To solve this problem, we need to understand the properties of logarithms and how to convert them into exponential form.

Logarithmic Properties

A logarithm is the inverse operation of exponentiation. In other words, if $y = \log _x a$ , then $x^y = a$ . This property is the foundation of logarithmic equations and is essential to solving this problem.

Converting Logarithmic to Exponential Form

To convert the given logarithmic equation $\log _x 36 = 2$ into exponential form, we need to apply the property mentioned above. This means that if $y = \log _x a$ , then $x^y = a$ . In this case, $a = 36$ and $y = 2$ . Therefore, we can rewrite the equation as $x^2 = 36$ .

Analyzing the Options

Now that we have converted the logarithmic equation into exponential form, we can analyze the given options to find the equivalent equation.

Option A: $2^x = 36$

This option is not equivalent to the given logarithmic equation. The base of the exponential function is 2, whereas the base of the logarithmic function is x. Therefore, this option is not correct.

Option B: $x^2 - 36$

This option is not equivalent to the given logarithmic equation. The equation $x^2 - 36$ is a quadratic equation, whereas the given logarithmic equation is an exponential equation. Therefore, this option is not correct.

Option C: $36^x = 2$

This option is not equivalent to the given logarithmic equation. The base of the exponential function is 36, whereas the base of the logarithmic function is x. Therefore, this option is not correct.

Option D: $x^2 = 36^2$

This option is equivalent to the given logarithmic equation. We can rewrite the equation $x^2 = 36^2$ as $x^2 = 1296$ , which is equivalent to the equation $x^2 = 36$ that we obtained by converting the logarithmic equation into exponential form.

Conclusion

In conclusion, the equivalent equation to $\log _x 36 = 2$ is $x^2 = 36^2$ . This equation is obtained by converting the logarithmic equation into exponential form using the properties of logarithms.

Final Answer

The final answer is: $\boxed{D}$

Understanding Logarithmic Equations

Logarithmic equations are a fundamental concept in mathematics, and they have numerous applications in various fields such as science, engineering, and economics. In this article, we will answer some frequently asked questions about logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log _x a$ , then $x^y = a$ .

Q: How do I convert a logarithmic equation into exponential form?

A: To convert a logarithmic equation into exponential form, you need to apply the property mentioned above. This means that if $y = \log _x a$ , then $x^y = a$ . For example, if we have the equation $\log _x 36 = 2$ , we can rewrite it as $x^2 = 36$ .

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

A: A logarithmic equation involves a logarithm, whereas an exponential equation involves an exponent. In other words, a logarithmic equation is in the form $y = \log _x a$ , whereas an exponential equation is in the form $x^y = a$ .

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to apply the properties of logarithms and convert the equation into exponential form. Once you have converted the equation into exponential form, you can solve for the variable.

Q: What are some common logarithmic equations?

A: Some common logarithmic equations include:

$\log _x a = b$
$\log _x a = \log _x b$
$\log _x (a \cdot b) = \log _x a + \log _x b$
$\log _x (a / b) = \log _x a - \log _x b$

Q: How do I use a calculator to solve a logarithmic equation?

A: To use a calculator to solve a logarithmic equation, you need to enter the equation in the correct format. For example, if you have the equation $\log _x 36 = 2$ , you can enter it as "log(36, x) = 2" and solve for x.

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

A: Logarithmic equations have numerous real-world applications in various fields such as science, engineering, and economics. Some examples include:

Calculating the pH of a solution
Determining the magnitude of an earthquake
Modeling population growth
Analyzing financial data

Conclusion

In conclusion, logarithmic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the properties of logarithms and how to convert logarithmic equations into exponential form, you can solve a wide range of problems involving logarithmic equations.

Final Answer

The final answer is: There is no final numerical answer to this article, as it is a Q&A article.