Select The Correct Answer.Which Exponential Equation Is Equivalent To This Logarithmic Equation? Log ⁡ X 5 + Log ⁡ X 12 = 7 \log_x 5 + \log_x 12 = 7 Lo G X ​ 5 + Lo G X ​ 12 = 7 A. X 7 = 60 X^7 = 60 X 7 = 60 B. 7 X = 60 7^x = 60 7 X = 60 C. X 7 = 17 X^7 = 17 X 7 = 17 D. 7 X = 17 7^x = 17 7 X = 17

Understanding the Basics of Exponential and Logarithmic Equations

Exponential and logarithmic equations are fundamental concepts in mathematics that are used to solve various problems in fields such as finance, science, and engineering. In this article, we will focus on solving exponential and logarithmic equations, with a specific emphasis on identifying the correct equivalent equation.

The Given Logarithmic Equation

The given logarithmic equation is $\log_x 5 + \log_x 12 = 7$. To solve this equation, we need to apply the properties of logarithms.

Applying the Properties of Logarithms

One of the key properties of logarithms is the product rule, which states that $\log_a b + \log_a c = \log_a (b \cdot c)$. We can apply this property to the given equation:

$$\log_x 5 + \log_x 12 = \log_x (5 \cdot 12) = \log_x 60$$

Simplifying the Equation

Now that we have applied the product rule, we can simplify the equation by equating the logarithmic expression to the given value:

$$\log_x 60 = 7$$

Converting the Logarithmic Equation to Exponential Form

To convert the logarithmic equation to exponential form, we need to apply the definition of logarithms, which states that $\log_a b = c$ is equivalent to $a^c = b$. In this case, we have:

$$x^7 = 60$$

Evaluating the Options

Now that we have converted the logarithmic equation to exponential form, we can evaluate the options:

A. $x^7 = 60$ B. $7^x = 60$ C. $x^7 = 17$ D. $7^x = 17$

Selecting the Correct Answer

Based on our analysis, we can see that option A is the correct answer. The exponential equation $x^7 = 60$ is equivalent to the given logarithmic equation $\log_x 5 + \log_x 12 = 7$.

Conclusion

In this article, we have demonstrated how to solve exponential and logarithmic equations by applying the properties of logarithms and converting the logarithmic equation to exponential form. We have also evaluated the options and selected the correct answer. This article provides a step-by-step guide to solving exponential and logarithmic equations, making it a valuable resource for students and professionals alike.

Common Mistakes to Avoid

When solving exponential and logarithmic equations, there are several common mistakes to avoid:

• Not applying the properties of logarithms: Failing to apply the product rule or other properties of logarithms can lead to incorrect solutions.
• Not converting the logarithmic equation to exponential form: Failing to convert the logarithmic equation to exponential form can make it difficult to evaluate the options.
• Not evaluating the options carefully: Failing to evaluate the options carefully can lead to selecting the incorrect answer.

Real-World Applications

Exponential and logarithmic equations have numerous real-world applications, including:

• Finance: Exponential and logarithmic equations are used to calculate interest rates, investment returns, and other financial metrics.
• Science: Exponential and logarithmic equations are used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Exponential and logarithmic equations are used to design and optimize systems, including electrical circuits, mechanical systems, and communication networks.

Tips and Tricks

When solving exponential and logarithmic equations, here are some tips and tricks to keep in mind:

• Use the properties of logarithms: Applying the product rule and other properties of logarithms can simplify the equation and make it easier to solve.
• Convert the logarithmic equation to exponential form: Converting the logarithmic equation to exponential form can make it easier to evaluate the options.
• Evaluate the options carefully: Evaluating the options carefully can help you select the correct answer.

Conclusion

In conclusion, solving exponential and logarithmic equations requires a deep understanding of the properties of logarithms and the ability to convert logarithmic equations to exponential form. By applying the product rule and other properties of logarithms, we can simplify the equation and make it easier to solve. Additionally, by converting the logarithmic equation to exponential form, we can evaluate the options and select the correct answer. This article provides a step-by-step guide to solving exponential and logarithmic equations, making it a valuable resource for students and professionals alike.

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about exponential and logarithmic equations.

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

A: An exponential equation is an equation that involves an exponential expression, such as $x^2 = 4$. A logarithmic equation is an equation that involves a logarithmic expression, such as $\log_x 5 = 2$.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential expression. For example, if you have the equation $x^2 = 4$, you can take the square root of both sides to get $x = \pm 2$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to apply the properties of logarithms. For example, if you have the equation $\log_x 5 = 2$, you can rewrite it as $x^2 = 5$.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log_a b + \log_a c = \log_a (b \cdot c)$. This means that you can combine two logarithmic expressions with the same base into a single logarithmic expression.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log_a b^c = c \log_a b$. This means that you can raise a logarithmic expression to a power by multiplying the exponent by the logarithmic expression.

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

A: To convert a logarithmic equation to exponential form, you need to apply the definition of logarithms. For example, if you have the equation $\log_x 5 = 2$, you can rewrite it as $x^2 = 5$.

Q: How do I evaluate the options in an exponential or logarithmic equation?

A: To evaluate the options in an exponential or logarithmic equation, you need to substitute each option into the equation and check if it is true. For example, if you have the equation $x^2 = 4$ and the options $x = 2$ and $x = -2$, you can substitute each option into the equation and check if it is true.

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

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

• Not applying the properties of logarithms
• Not converting the logarithmic equation to exponential form
• Not evaluating the options carefully

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

A: Exponential and logarithmic equations have numerous real-world applications, including:

• Finance: Exponential and logarithmic equations are used to calculate interest rates, investment returns, and other financial metrics.
• Science: Exponential and logarithmic equations are used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Exponential and logarithmic equations are used to design and optimize systems, including electrical circuits, mechanical systems, and communication networks.

Q: What are some tips and tricks for solving exponential and logarithmic equations?

A: Some tips and tricks for solving exponential and logarithmic equations include:

• Using the properties of logarithms to simplify the equation
• Converting the logarithmic equation to exponential form to make it easier to evaluate the options
• Evaluating the options carefully to select the correct answer

Conclusion

In conclusion, exponential and logarithmic equations are fundamental concepts in mathematics that have numerous real-world applications. By understanding the properties of logarithms and how to convert logarithmic equations to exponential form, you can solve exponential and logarithmic equations with ease. Additionally, by following the tips and tricks outlined in this article, you can avoid common mistakes and select the correct answer with confidence.