Select The Correct Answer.What Is The Value Of $x$? Log ⁡ 2 ( X + 7 ) = 2 \log _2(x+7)=2 Lo G 2 ​ ( X + 7 ) = 2 A. 11 B. 250 C. 22 D. -3

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, $\log _2(x+7)=2$, and determine the value of $x$. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In the given equation, $\log _2(x+7)=2$, the base of the logarithm is 2, and the argument of the logarithm is $x+7$. The equation states that the logarithm of $x+7$ with base 2 is equal to 2.

Step 1: Rewrite the Equation in Exponential Form

To solve the logarithmic equation, we can rewrite it in exponential form. The exponential form of the equation is $2^2 = x + 7$. This step is based on the property of logarithms that states $\log_b(a) = c$ is equivalent to $b^c = a$.

Step 2: Simplify the Exponential Form

The exponential form of the equation is $2^2 = x + 7$. We can simplify this equation by evaluating the exponent. $2^2 = 4$, so the equation becomes $4 = x + 7$.

Step 3: Isolate the Variable

To isolate the variable $x$, we need to get rid of the constant term on the same side of the equation. We can do this by subtracting 7 from both sides of the equation. This gives us $x = 4 - 7$.

Step 4: Simplify the Expression

The expression $4 - 7$ can be simplified by evaluating the subtraction. This gives us $x = -3$.

Conclusion

In this article, we solved the logarithmic equation $\log _2(x+7)=2$ and determined the value of $x$. We broke down the solution into manageable steps, making it easy to follow and understand. The final answer is $x = -3$.

Answer

The correct answer is:

• D. -3

Discussion

This problem is a great example of how to solve logarithmic equations. The key concept is to rewrite the equation in exponential form and then simplify the expression. This problem requires a clear understanding of the properties of logarithms and the ability to apply them to solve the equation.

Related Problems

If you want to practice solving logarithmic equations, here are some related problems:

• $\log _3(x-2)=1$
• $\log _5(x+1)=3$
• $\log _2(x-3)=4$

These problems require the same steps as the original problem, and they will help you practice solving logarithmic equations.

Final Thoughts

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. It is an equation that states that the logarithm of a number with a certain base is equal to a certain value.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to rewrite it in exponential form and then simplify the expression. This involves using the property of logarithms that states $\log_b(a) = c$ is equivalent to $b^c = a$.

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, $\log _2(x+7)=2$ is a logarithmic equation, while $2^2 = x + 7$ is an exponential equation.

Q: How do I determine the base of a logarithmic equation?

A: The base of a logarithmic equation is the number that is used as the base of the logarithm. For example, in the equation $\log _2(x+7)=2$, the base is 2.

Q: What is the argument of a logarithmic equation?

A: The argument of a logarithmic equation is the number that is being logged. For example, in the equation $\log _2(x+7)=2$, the argument is $x+7$.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you need to rewrite it in exponential form and then simplify the expression. This involves using the property of logarithms that states $\log_b(a) = c$ is equivalent to $b^c = a$.

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

A: A logarithmic equation is an equation that involves a logarithm, while an algebraic equation is an equation that involves variables and constants. For example, $\log _2(x+7)=2$ is a logarithmic equation, while $x + 7 = 4$ is an algebraic equation.

Q: How do I solve a logarithmic equation with a negative base?

A: To solve a logarithmic equation with a negative base, you need to rewrite it in exponential form and then simplify the expression. This involves using the property of logarithms that states $\log_b(a) = c$ is equivalent to $b^c = a$. However, you need to be careful when working with negative bases, as the result may be a complex number.

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

A: A logarithmic equation is an equation that involves a logarithm, while a trigonometric equation is an equation that involves trigonometric functions such as sine, cosine, and tangent. For example, $\log _2(x+7)=2$ is a logarithmic equation, while $\sin(x) = 0.5$ is a trigonometric equation.

Q: How do I solve a logarithmic equation with a fractional base?

A: To solve a logarithmic equation with a fractional base, you need to rewrite it in exponential form and then simplify the expression. This involves using the property of logarithms that states $\log_b(a) = c$ is equivalent to $b^c = a$. However, you need to be careful when working with fractional bases, as the result may be a complex number.

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

A: A logarithmic equation is an equation that involves a logarithm, while a polynomial equation is an equation that involves variables and constants raised to various powers. For example, $\log _2(x+7)=2$ is a logarithmic equation, while $x^2 + 7x + 12 = 0$ is a polynomial equation.

Q: How do I solve a logarithmic equation with a logarithm of a logarithm?

A: To solve a logarithmic equation with a logarithm of a logarithm, you need to rewrite it in exponential form and then simplify the expression. This involves using the property of logarithms that states $\log_b(a) = c$ is equivalent to $b^c = a$. However, you need to be careful when working with logarithms of logarithms, as the result may be a complex number.

Conclusion

In this article, we have answered some of the most frequently asked questions about logarithmic equations. We have covered topics such as the definition of a logarithmic equation, how to solve a logarithmic equation, and the difference between a logarithmic equation and an exponential equation. We hope that this article has been helpful in answering your questions and providing you with a better understanding of logarithmic equations.