Select The Correct Answer.Which Exponential Equation Is Equivalent To This Logarithmic Equation $\log X = 4$?A. $10^4 = X$B. $4^{20} = 2$C. $10^2 = 4$D. $4 \cdot X = 10$

Introduction

Logarithmic and exponential equations are fundamental concepts in mathematics, and understanding how to solve them is crucial for success in various fields, including science, engineering, and finance. In this article, we will focus on solving exponential equations that are equivalent to logarithmic equations. Specifically, we will explore the correct answer to the question: Which exponential equation is equivalent to the logarithmic equation $\log x = 4$?

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, a logarithm answers the question: "To what power must a base be raised to obtain a given number?" For example, the logarithmic equation $\log x = 4$ can be rewritten as $x = 10^4$, where $10$ is the base and $4$ is the exponent.

Converting Logarithmic Equations to Exponential Equations

To convert a logarithmic equation to an exponential equation, we need to use the definition of a logarithm. Specifically, if $\log x = y$, then $x = a^y$, where $a$ is the base of the logarithm. In the case of the logarithmic equation $\log x = 4$, we can rewrite it as $x = 10^4$, where $10$ is the base and $4$ is the exponent.

Analyzing the Options

Now that we have a clear understanding of how to convert logarithmic equations to exponential equations, let's analyze the options:

A. $10^4 = x$ B. $4^{20} = 2$ C. $10^2 = 4$ D. $4 \cdot x = 10$

Option A: $10^4 = x$

This option is equivalent to the logarithmic equation $\log x = 4$. As we discussed earlier, the logarithmic equation $\log x = 4$ can be rewritten as $x = 10^4$. Therefore, option A is the correct answer.

Option B: $4^{20} = 2$

This option is not equivalent to the logarithmic equation $\log x = 4$. The base of the logarithm is $10$, not $4$. Therefore, option B is incorrect.

Option C: $10^2 = 4$

This option is not equivalent to the logarithmic equation $\log x = 4$. The exponent is $2$, not $4$. Therefore, option C is incorrect.

Option D: $4 \cdot x = 10$

This option is not equivalent to the logarithmic equation $\log x = 4$. The equation involves a multiplication, not an exponentiation. Therefore, option D is incorrect.

Conclusion

In conclusion, the correct answer to the question: Which exponential equation is equivalent to the logarithmic equation $\log x = 4$? is option A: $10^4 = x$. This option is equivalent to the logarithmic equation $\log x = 4$ because it involves the same base and exponent. We hope this article has provided a clear understanding of how to solve exponential and logarithmic equations and has helped you to identify the correct answer.

Common Mistakes to Avoid

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

• Incorrect base: Make sure to use the correct base when converting a logarithmic equation to an exponential equation.
• Incorrect exponent: Make sure to use the correct exponent when converting a logarithmic equation to an exponential equation.
• Incorrect operation: Make sure to use the correct operation when solving an exponential or logarithmic equation.

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 computer networks.

Final Thoughts

In conclusion, solving exponential and logarithmic equations is a crucial skill that has numerous real-world applications. By understanding how to convert logarithmic equations to exponential equations, we can solve a wide range of problems in finance, science, and engineering. We hope this article has provided a clear understanding of how to solve exponential and logarithmic equations and has helped you to identify the correct answer.