What Is The Solution To The Equation Below? Round Your Answer To Two Decimal Places.$\log _4 X=2.1$A. $x=8.17$B. $x=19.45$C. $x=8.40$D. $x=18.38$

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of logarithms. In this article, we will focus on solving a specific logarithmic equation, $\log _4 x=2.1$, and explore the different methods to find the solution.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. The logarithm of a number $x$ to a base $b$ is denoted by $\log_b x$. In the given equation, $\log _4 x=2.1$, we are looking for the value of $x$ that satisfies this equation.

The Properties of Logarithms

Before we dive into solving the equation, it's essential to understand the properties of logarithms. The two main properties of logarithms are:

• The power property: $\log_b (x^y) = y \log_b x$
• The product property: $\log_b (xy) = \log_b x + \log_b y$

These properties will be useful in solving the equation.

Solving the Equation

To solve the equation $\log _4 x=2.1$, we can use the definition of a logarithm. The logarithm $\log_b x$ is defined as the exponent to which the base $b$ must be raised to obtain the number $x$. In other words, $\log_b x = y$ if and only if $b^y = x$.

Using this definition, we can rewrite the equation as:

$$4^{2.1} = x$$

Now, we can use a calculator to find the value of $4^{2.1}$.

Calculating the Value

Using a calculator, we find that:

$$4^{2.1} \approx 8.17$$

Therefore, the solution to the equation $\log _4 x=2.1$ is $x \approx 8.17$.

Rounding the Answer

The problem asks us to round the answer to two decimal places. Therefore, the final answer is $x = 8.17$.

Conclusion

In this article, we solved a logarithmic equation using the definition of a logarithm and a calculator. We also explored the properties of logarithms and how they can be used to simplify logarithmic equations. The solution to the equation $\log _4 x=2.1$ is $x \approx 8.17$.

Comparison with Other Options

Let's compare our solution with the other options:

• Option A: $x = 8.17$ (our solution)
• Option B: $x = 19.45$ (this is not a possible solution)
• Option C: $x = 8.40$ (this is not a possible solution)
• Option D: $x = 18.38$ (this is not a possible solution)

As we can see, our solution is the only possible solution to the equation.

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. The logarithm of a number $x$ to a base $b$ is denoted by $\log_b x$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the definition of a logarithm. The logarithm $\log_b x$ is defined as the exponent to which the base $b$ must be raised to obtain the number $x$. In other words, $\log_b x = y$ if and only if $b^y = x$.

Q: What are the properties of logarithms?

A: The two main properties of logarithms are:

• The power property: $\log_b (x^y) = y \log_b x$
• The product property: $\log_b (xy) = \log_b x + \log_b y$

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

A: To use a calculator to solve a logarithmic equation, you can enter the equation in the calculator and press the "solve" or "calculate" button. The calculator will then give you the solution to the equation.

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 $\log_4 x = 2.1$ is a logarithmic equation, while the equation $4^x = 8.17$ is an exponential equation.

Q: Can I use logarithmic equations to solve real-world problems?

A: Yes, logarithmic equations can be used to solve real-world problems. For example, logarithmic equations can be used to model population growth, chemical reactions, and financial transactions.

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

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

• Not using the correct base: Make sure to use the correct base when solving a logarithmic equation.
• Not using the correct exponent: Make sure to use the correct exponent when solving a logarithmic equation.
• Not checking the solution: Make sure to check the solution to the equation to ensure that it is correct.

Q: How do I check the solution to a logarithmic equation?

A: To check the solution to a logarithmic equation, you can plug the solution back into the original equation and check if it is true. If the solution is true, then it is the correct solution to the equation.

Q: What are some common applications of logarithmic equations?

A: Some common applications of logarithmic equations include:

• Population growth: Logarithmic equations can be used to model population growth and predict future population sizes.
• Chemical reactions: Logarithmic equations can be used to model chemical reactions and predict the rates of reaction.
• Financial transactions: Logarithmic equations can be used to model financial transactions and predict future stock prices.

Conclusion

In this article, we have answered some frequently asked questions about logarithmic equations. We have discussed the definition of a logarithmic equation, the properties of logarithms, and how to use a calculator to solve a logarithmic equation. We have also discussed some common mistakes to avoid when solving logarithmic equations and how to check the solution to a logarithmic equation.