What Is The Value Of $x$ In $\log_2(x) - 3 = 1$?A. $2^1$ B. $2^3$ C. $2^2$ D. $2^4$

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Introduction

In this article, we will explore the concept of logarithms and how to solve equations involving logarithms. We will focus on the given equation $\log_2(x) - 3 = 1$ and determine the value of $x$. This equation involves a logarithmic function with base 2, and we will use properties of logarithms to solve for $x$.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. In other words, if $y = \log_b(x)$, then $b^y = x$. The logarithm of a number $x$ with base $b$ is the exponent to which $b$ must be raised to produce $x$. For example, $\log_2(8) = 3$ because $2^3 = 8$.

Solving the Equation

To solve the equation $\log_2(x) - 3 = 1$, we need to isolate the logarithmic term. We can do this by adding 3 to both sides of the equation:

$$\log_2(x) - 3 + 3 = 1 + 3$$

This simplifies to:

$$\log_2(x) = 4$$

Using Properties of Logarithms

Now that we have isolated the logarithmic term, we can use the definition of a logarithm to rewrite the equation in exponential form:

$$2^4 = x$$

This means that $x$ is equal to $2^4$, which is equal to 16.

Conclusion

In this article, we have solved the equation $\log_2(x) - 3 = 1$ and determined the value of $x$. We used properties of logarithms to isolate the logarithmic term and then used the definition of a logarithm to rewrite the equation in exponential form. The final answer is $x = 2^4$, which is equal to 16.

Final Answer

The final answer is $\boxed{16}$.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Start with the given equation: $\log_2(x) - 3 = 1$
2. Add 3 to both sides of the equation: $\log_2(x) - 3 + 3 = 1 + 3$
3. Simplify the equation: $\log_2(x) = 4$
4. Use the definition of a logarithm to rewrite the equation in exponential form: $2^4 = x$
5. Evaluate the expression: $x = 2^4 = 16$

Common Mistakes

Here are some common mistakes to avoid when solving this problem:

• Not isolating the logarithmic term
• Not using the definition of a logarithm to rewrite the equation in exponential form
• Not evaluating the expression correctly

Tips and Tricks

Here are some tips and tricks to help you solve this problem:

• Make sure to isolate the logarithmic term before using the definition of a logarithm
• Use the definition of a logarithm to rewrite the equation in exponential form
• Evaluate the expression correctly

Real-World Applications

This problem has real-world applications in fields such as computer science, engineering, and economics. For example, logarithmic functions are used to model population growth, financial transactions, and signal processing.

Conclusion

In conclusion, solving the equation $\log_2(x) - 3 = 1$ requires using properties of logarithms and the definition of a logarithm. By following the step-by-step solution and avoiding common mistakes, you can determine the value of $x$ and apply the concept to real-world problems.

Final Thoughts

Logarithmic functions are an essential part of mathematics, and solving equations involving logarithms is a critical skill to develop. By mastering this skill, you can apply it to a wide range of problems and fields, from computer science to economics.

Introduction

In the previous article, we explored the concept of logarithmic equations and solved the equation $\log_2(x) - 3 = 1$. In this article, we will answer some frequently asked questions (FAQs) about logarithmic equations.

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

A: A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. An exponential equation, on the other hand, is an equation that involves an exponential function. For example, $\log_2(x) = 4$ is a logarithmic equation, while $2^x = 16$ is an exponential equation.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic term and then use the definition of a logarithm to rewrite the equation in exponential form. For example, to solve the equation $\log_2(x) - 3 = 1$, you would add 3 to both sides of the equation and then use the definition of a logarithm to rewrite the equation in exponential form.

Q: What is the base of a logarithmic function?

A: The base of a logarithmic function is the number that is raised to a power to produce the input value. For example, in the equation $\log_2(x) = 4$, the base is 2.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to use the definition of a logarithm to rewrite the expression in exponential form. For example, to evaluate the expression $\log_2(16)$, you would rewrite it as $2^4$.

Q: What is the difference between a common logarithm and a natural logarithm?

A: A common logarithm is a logarithm with base 10, while a natural logarithm is a logarithm with base e. For example, $\log_{10}(x)$ is a common logarithm, while $\ln(x)$ is a natural logarithm.

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 calculator and then use the logarithmic function to solve for the variable. For example, to solve the equation $\log_2(x) = 4$ using a calculator, you would enter the equation and then use the logarithmic function to solve for x.

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

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

• Not isolating the logarithmic term
• Not using the definition of a logarithm to rewrite the equation in exponential form
• Not evaluating the expression correctly
• Using the wrong base for the logarithmic function

Q: How do I apply logarithmic equations to real-world problems?

A: Logarithmic equations can be applied to a wide range of real-world problems, including:

• Modeling population growth
• Analyzing financial transactions
• Signal processing
• Computer science

Conclusion

In conclusion, logarithmic equations are an essential part of mathematics, and solving them requires a deep understanding of the concept of logarithms. By mastering the skills and concepts presented in this article, you can apply logarithmic equations to a wide range of problems and fields.

Final Thoughts

Logarithmic equations are a powerful tool for solving problems in mathematics and other fields. By understanding the concept of logarithms and how to solve logarithmic equations, you can apply this knowledge to a wide range of problems and fields.

Additional Resources

For additional resources on logarithmic equations, including videos, tutorials, and practice problems, please visit the following websites:

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

Glossary

• Logarithmic equation: An equation that involves a logarithmic function.
• Exponential equation: An equation that involves an exponential function.
• Base: The number that is raised to a power to produce the input value.
• Common logarithm: A logarithm with base 10.
• Natural logarithm: A logarithm with base e.
• Logarithmic function: The inverse of an exponential function.