Rewrite the logarithmic equation in exponential form

Oct 13, 2025 by miodeya.com 88 views

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will explore a specific logarithmic equation and determine which of the given statements is true regarding its solution.

The Logarithmic Equation

The given logarithmic equation is:

{ \begin{aligned} \log _2(x+11) &= 4 \\ x+11 &= 2^4 \\ x+11 &= 16 \\ x &= 5 \end{aligned} \}

This equation involves a logarithm with base 2, and the goal is to solve for the variable x.

Understanding the Properties of Logarithms

Before we proceed with solving the equation, it's essential to understand the properties of logarithms. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In other words, if $\log_b(a) = c$ , then $b^c = a$ .

Solving the Logarithmic Equation

Now, let's solve the given logarithmic equation step by step.

Rewrite the logarithmic equation in exponential form: The first step in solving the logarithmic equation is to rewrite it in exponential form. This means that we need to express the logarithmic equation as an exponential equation.

{ \begin{aligned} \log _2(x+11) &= 4 \\ 2^4 &= x+11 \end{aligned} \}

Simplify the exponential equation: The next step is to simplify the exponential equation by evaluating the base raised to the power of the exponent.

{ \begin{aligned} 2^4 &= x+11 \\ 16 &= x+11 \end{aligned} \}

Isolate the variable x: To solve for the variable x, we need to isolate it on one side of the equation. This can be done by subtracting 11 from both sides of the equation.

{ \begin{aligned} 16 &= x+11 \\ 16-11 &= x \\ 5 &= x \end{aligned} \}

Evaluating the Solution

Now that we have solved the logarithmic equation, let's evaluate the solution. The solution to the equation is x = 5. However, we need to determine which of the given statements is true regarding this solution.

Statement A: $x=5$ is not a true solution because

Statement A claims that x = 5 is not a true solution because the original equation involves a logarithm with base 2, and the solution x = 5 does not satisfy the original equation.

Analysis of Statement A

To analyze statement A, let's substitute x = 5 into the original equation and see if it satisfies the equation.

{ \begin{aligned} \log _2(x+11) &= 4 \\ \log _2(5+11) &= 4 \\ \log _2(16) &= 4 \\ 4 &= 4 \end{aligned} \}

As we can see, the solution x = 5 satisfies the original equation. Therefore, statement A is false.

Conclusion

In conclusion, the solution to the logarithmic equation is x = 5. This solution satisfies the original equation, and therefore, statement A is false. The correct solution to the logarithmic equation is x = 5.

Final Thoughts

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we explored a specific logarithmic equation and determined which of the given statements is true regarding its solution. We found that the solution x = 5 satisfies the original equation, and therefore, statement A is false.

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will address some frequently asked questions (FAQs) about logarithmic equations.

Q1: What is a logarithmic equation?

A1: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $\log_b(a) = c$ , then $b^c = a$ .

Q2: How do I solve a logarithmic equation?

A2: To solve a logarithmic equation, you need to follow these steps:

Rewrite the logarithmic equation in exponential form.
Simplify the exponential equation by evaluating the base raised to the power of the exponent.
Isolate the variable on one side of the equation.

Q3: What is the difference between a logarithmic equation and an exponential equation?

A3: A logarithmic equation and an exponential equation are inverse operations. A logarithmic equation is in the form $\log_b(a) = c$ , while an exponential equation is in the form $b^c = a$ .

Q4: How do I determine the base of a logarithmic equation?

A4: The base of a logarithmic equation is the number that is raised to the power of the exponent. For example, in the equation $\log_2(x+11) = 4$ , the base is 2.

Q5: Can a logarithmic equation have multiple solutions?

A5: Yes, a logarithmic equation can have multiple solutions. This occurs when the equation has multiple values that satisfy the equation.

Q6: How do I graph a logarithmic equation?

A6: To graph a logarithmic equation, you need to follow these steps:

Rewrite the logarithmic equation in exponential form.
Graph the exponential equation.
Reflect the graph of the exponential equation across the y-axis to obtain the graph of the logarithmic equation.

Q7: What is the domain of a logarithmic equation?

A7: The domain of a logarithmic equation is the set of all possible values of the variable that satisfy the equation. For example, in the equation $\log_2(x+11) = 4$ , the domain is $x > -11$ .

Q8: Can a logarithmic equation have a negative solution?

A8: Yes, a logarithmic equation can have a negative solution. This occurs when the base of the logarithm is negative.

Q9: How do I solve a logarithmic equation with a negative base?

A9: To solve a logarithmic equation with a negative base, you need to follow these steps:

Rewrite the logarithmic equation in exponential form.
Simplify the exponential equation by evaluating the base raised to the power of the exponent.
Isolate the variable on one side of the equation.

Q10: What is the range of a logarithmic equation?

A10: The range of a logarithmic equation is the set of all possible values of the logarithm. For example, in the equation $\log_2(x+11) = 4$ , the range is all real numbers.

Conclusion

In conclusion, logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. We have addressed some frequently asked questions (FAQs) about logarithmic equations, including how to solve them, how to determine the base, and how to graph them.

Final Thoughts

Logarithmic equations are a powerful tool in mathematics, and understanding them is essential for solving a wide range of problems. We hope that this article has provided you with a better understanding of logarithmic equations and how to solve them.

Additional Resources

If you are looking for additional resources on logarithmic equations, we recommend the following:

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

These resources provide a wealth of information on logarithmic equations, including tutorials, examples, and practice problems.