Type The Correct Answer In The Box. Use Numerals Instead Of Words.What Value Of $x$ Satisfies This Equation? Log ⁡ ( 2 X ) = 2 \log (2x) = 2 Lo G ( 2 X ) = 2 The Value Of $x$ Is □ \square □ .

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific type of logarithmic equation, namely the equation $\log (2x) = 2$. Our goal is to find the value of $x$ that satisfies this equation.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's take a moment to understand what logarithmic equations are. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log_b(x)$, then $b^y = x$. This means that logarithmic equations can be rewritten in exponential form.

Rewriting the Equation in Exponential Form

To solve the equation $\log (2x) = 2$, we can rewrite it in exponential form. Since the base of the logarithm is not specified, we will assume that it is the common logarithm, which has a base of 10. Therefore, we can rewrite the equation as:

$$10^2 = 2x$$

Simplifying the Equation

Now that we have rewritten the equation in exponential form, we can simplify it by evaluating the exponent. In this case, $10^2 = 100$. Therefore, we can rewrite the equation as:

$$100 = 2x$$

Solving for $x$

Now that we have simplified the equation, we can solve for $x$. To do this, we can divide both sides of the equation by 2, which gives us:

$$x = \frac{100}{2}$$

Evaluating the Expression

Finally, we can evaluate the expression on the right-hand side of the equation. Dividing 100 by 2 gives us 50. Therefore, the value of $x$ that satisfies the equation is:

$$x = 50$$

Conclusion

In this article, we have solved a logarithmic equation using the properties of logarithms and exponentiation. We have rewritten the equation in exponential form, simplified it, and solved for $x$. The value of $x$ that satisfies the equation is 50.

Tips and Tricks

Here are some tips and tricks to help you solve logarithmic equations:

• Use the properties of logarithms: Logarithmic equations can be rewritten in exponential form using the properties of logarithms.
• Simplify the equation: Simplify the equation by evaluating the exponent and combining like terms.
• Solve for the variable: Solve for the variable by isolating it on one side of the equation.
• Check your answer: Check your answer by plugging it back into the original equation.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving logarithmic equations:

• Not rewriting the equation in exponential form: Failing to rewrite the equation in exponential form can make it difficult to solve.
• Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
• Not solving for the variable: Failing to solve for the variable can make it difficult to find the solution.
• Not checking your answer: Failing to check your answer can lead to incorrect solutions.

Real-World Applications

Logarithmic equations have many real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design and optimize systems.

Conclusion

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, if $y = \log_b(x)$, then $b^y = x$. This means that logarithmic equations can be rewritten in exponential form.

Q: How do I rewrite a logarithmic equation in exponential form?

A: To rewrite a logarithmic equation in exponential form, you can use the property of logarithms that states $\log_b(x) = y$ is equivalent to $b^y = x$. For example, if we have the equation $\log (2x) = 2$, we can rewrite it in exponential form as $10^2 = 2x$.

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

A: A logarithmic equation and an exponential equation are related but distinct concepts. A logarithmic equation involves a logarithm, which is the inverse operation of exponentiation, while an exponential equation involves an exponent. For example, the equation $\log (2x) = 2$ is a logarithmic equation, while the equation $2^x = 10$ is an exponential equation.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can follow these steps:

1. Rewrite the equation in exponential form: Use the property of logarithms to rewrite the equation in exponential form.
2. Simplify the equation: Simplify the equation by evaluating the exponent and combining like terms.
3. Solve for the variable: Solve for the variable by isolating it on one side of the equation.
4. Check your answer: Check your answer by plugging it back into the original equation.

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

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

• Not rewriting the equation in exponential form: Failing to rewrite the equation in exponential form can make it difficult to solve.
• Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
• Not solving for the variable: Failing to solve for the variable can make it difficult to find the solution.
• Not checking your answer: Failing to check your answer can lead to incorrect solutions.

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

A: Logarithmic equations have many real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design and optimize systems.

Q: How can I practice solving logarithmic equations?

A: You can practice solving logarithmic equations by working through examples and exercises in a textbook or online resource. You can also try solving logarithmic equations on your own by using the steps outlined above.

Q: What are some advanced topics in logarithmic equations?

A: Some advanced topics in logarithmic equations include:

• Logarithmic differentiation: This involves using logarithmic equations to differentiate functions.
• Logarithmic integration: This involves using logarithmic equations to integrate functions.
• Logarithmic series: This involves using logarithmic equations to represent series.

Conclusion

In conclusion, logarithmic equations are an important topic in mathematics that have many real-world applications. By following the steps outlined above and practicing solving logarithmic equations, you can become proficient in solving these equations and apply them to real-world problems.