Which Equation Is Equivalent To $\log _x 36 = 2$?A. $2^x = 36$B. $x^2 - 36$C. $36^x = 2$D. $x^2 = 36^2$

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will explore the concept of logarithmic equations and provide a step-by-step guide on how to solve them. We will also examine a specific equation, $\log _x 36 = 2$, and determine which of the given options is equivalent to it.

What are Logarithmic Equations?

A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, a logarithmic equation is an equation that can be rewritten in exponential form. For example, the equation $\log _x 36 = 2$ can be rewritten as $x^2 = 36$.

Properties of Logarithms

To solve logarithmic equations, it is essential to understand the properties of logarithms. The two main properties of logarithms are:

• Product Property: $\log _a (bc) = \log _a b + \log _a c$
• Power Property: $\log _a b^c = c \log _a b$

Solving Logarithmic Equations

To solve a logarithmic equation, we need to isolate the logarithmic term. We can do this by using the properties of logarithms. Here are the steps to solve a logarithmic equation:

1. Rewrite the equation in exponential form: Rewrite the logarithmic equation in exponential form using the property $\log _a b = c \implies a^c = b$.
2. Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary variables.
3. Solve for the variable: Solve for the variable by isolating it on one side of the equation.

Solving the Equation $\log _x 36 = 2$

Now, let's apply the steps above to solve the equation $\log _x 36 = 2$. We can rewrite this equation in exponential form as follows:

$$x^2 = 36$$

This equation can be rewritten as:

$$x^2 - 36 = 0$$

We can factor the left-hand side of the equation as follows:

$$(x - 6)(x + 6) = 0$$

This equation has two solutions: $x = 6$ and $x = -6$.

Which Option is Equivalent to the Equation?

Now, let's examine the given options and determine which one is equivalent to the equation $\log _x 36 = 2$.

• Option A: $2^x = 36$
• Option B: $x^2 - 36$
• Option C: $36^x = 2$
• Option D: $x^2 = 36^2$

We can see that option D is equivalent to the equation $\log _x 36 = 2$. This is because $x^2 = 36^2$ is equivalent to $x^2 = 36$, which is the same as the equation $\log _x 36 = 2$.

Conclusion

In conclusion, solving logarithmic equations requires a deep understanding of the properties of logarithms. By following the steps outlined above, we can solve logarithmic equations and determine which option is equivalent to the equation. In this article, we solved the equation $\log _x 36 = 2$ and determined that option D is equivalent to it.

Final Answer

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, a logarithmic equation is an equation that can be rewritten in exponential form.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic term. You can do this by using the properties of logarithms. Here are the steps to solve a logarithmic equation:

1. Rewrite the equation in exponential form: Rewrite the logarithmic equation in exponential form using the property $\log _a b = c \implies a^c = b$.
2. Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary variables.
3. Solve for the variable: Solve for the variable by isolating it on one side of the equation.

Q: What are the properties of logarithms?

A: The two main properties of logarithms are:

• Product Property: $\log _a (bc) = \log _a b + \log _a c$
• Power Property: $\log _a b^c = c \log _a b$

Q: How do I use the product property to solve a logarithmic equation?

A: To use the product property to solve a logarithmic equation, you need to rewrite the equation in a way that allows you to apply the product property. For example, if you have the equation $\log _a (bc) = \log _a b + \log _a c$, you can rewrite it as $a^{\log _a (bc)} = a^{\log _a b + \log _a c}$.

Q: How do I use the power property to solve a logarithmic equation?

A: To use the power property to solve a logarithmic equation, you need to rewrite the equation in a way that allows you to apply the power property. For example, if you have the equation $\log _a b^c = c \log _a b$, you can rewrite it as $a^{\log _a b^c} = a^{c \log _a b}$.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, the equation $\log _a b = c$ is a logarithmic equation, while the equation $a^c = b$ is an exponential equation.

Q: How do I determine which option is equivalent to a logarithmic equation?

A: To determine which option is equivalent to a logarithmic equation, you need to rewrite the equation in exponential form and compare it to the options. For example, if you have the equation $\log _x 36 = 2$, you can rewrite it as $x^2 = 36$ and compare it to the options.

Q: What is the final answer to the equation $\log _x 36 = 2$?

A: The final answer to the equation $\log _x 36 = 2$ is option D: $x^2 = 36^2$.

Conclusion

In conclusion, logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. By following the steps outlined above, you can solve logarithmic equations and determine which option is equivalent to the equation.