Select The Correct Answer.What Is The Solution Set To This Equation? Log ⁡ 4 ( X + 3 ) + Log ⁡ 4 X = 1 \log _4(x+3)+\log _4 X=1 Lo G 4 ​ ( X + 3 ) + Lo G 4 ​ X = 1 A. X = 1 X=1 X = 1 B. X = − 3 X=-3 X = − 3 And X = 0 X=0 X = 0 C. X = 0 X=0 X = 0 D. X = 1 X=1 X = 1 And X = − 4 X=-4 X = − 4

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore the solution set to the equation $\log _4(x+3)+\log _4 x=1$. We will break down the solution process into manageable steps, making it easy to understand and follow along.

Understanding Logarithmic Properties

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

• Product Property: $\log _a (xy) = \log _a x + \log _a y$
• Power Property: $\log _a x^y = y \log _a x$

These properties will help us simplify the equation and make it easier to solve.

Simplifying the Equation

Let's start by simplifying the equation using the product property:

$$\log _4(x+3)+\log _4 x=1$$

Using the product property, we can rewrite the equation as:

$$\log _4((x+3)x)=1$$

This simplifies to:

$$\log _4(x^2+3x)=1$$

Using the Definition of Logarithms

The definition of logarithms states that if $y = \log _a x$, then $a^y = x$. We can use this definition to rewrite the equation:

$$4^1=x^2+3x$$

This simplifies to:

$$4=x^2+3x$$

Rearranging the Equation

To make it easier to solve, let's rearrange the equation by subtracting 4 from both sides:

$$0=x^2+3x-4$$

This is a quadratic equation in the form $ax^2+bx+c=0$, where $a=1$, $b=3$, and $c=-4$.

Solving the Quadratic Equation

We can solve the quadratic equation using the quadratic formula:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Plugging in the values, we get:

$$x=\frac{-3\pm\sqrt{3^2-4(1)(-4)}}{2(1)}$$

Simplifying, we get:

$$x=\frac{-3\pm\sqrt{25}}{2}$$

This simplifies to:

$$x=\frac{-3\pm5}{2}$$

Finding the Solutions

We have two possible solutions:

$$x_1=\frac{-3+5}{2}=\frac{2}{2}=1$$

$$x_2=\frac{-3-5}{2}=\frac{-8}{2}=-4$$

Conclusion

In conclusion, the solution set to the equation $\log _4(x+3)+\log _4 x=1$ is $x=1$ and $x=-4$. We used the product property and the definition of logarithms to simplify the equation, and then solved the resulting quadratic equation using the quadratic formula.

Answer

The correct answer is:

• D. $x=1$ and $x=-4$

Final Thoughts

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore some common questions and answers related to logarithmic equations.

Q: What is the definition of a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, it is an equation that involves a variable in the exponent of a number.

Q: What are the two main properties of logarithms?

A: The two main properties of logarithms are:

• Product Property: $\log _a (xy) = \log _a x + \log _a y$
• Power Property: $\log _a x^y = y \log _a x$

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can use the product property and the definition of logarithms. For example, if you have the equation $\log _a (xy)$, you can rewrite it as $\log _a x + \log _a y$ using the product property.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the definition of logarithms and the quadratic formula. For example, if you have the equation $\log _a x = b$, you can rewrite it as $a^b = x$ using the definition of logarithms. Then, you can use the quadratic formula to solve for $x$.

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 $\log _a x = b$ is a logarithmic equation, while the equation $a^x = b$ is an exponential equation.

Q: Can you give an example of a logarithmic equation?

A: Yes, here is an example of a logarithmic equation:

$$\log _4(x+3)+\log _4 x=1$$

This equation can be solved using the product property and the definition of logarithms.

Q: Can you give an example of a quadratic equation that arises from a logarithmic equation?

A: Yes, here is an example of a quadratic equation that arises from a logarithmic equation:

$$x^2+3x-4=0$$

This equation can be solved using the quadratic formula.

Q: What is the solution set to the equation $\log _4(x+3)+\log _4 x=1$?

A: The solution set to the equation $\log _4(x+3)+\log _4 x=1$ is $x=1$ and $x=-4$.

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

A: To determine the domain of a logarithmic equation, you need to make sure that the argument of the logarithm is positive. For example, if you have the equation $\log _a x$, you need to make sure that $x$ is positive.

Q: Can you give an example of a logarithmic equation with a base of 10?

A: Yes, here is an example of a logarithmic equation with a base of 10:

$$\log _{10} x = 2$$

This equation can be rewritten as $10^2 = x$ using the definition of logarithms.

Conclusion

In conclusion, logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. By understanding the properties of logarithms and using the definition of logarithms, you can simplify and solve logarithmic equations. We hope this Q&A guide has been helpful in answering your questions about logarithmic equations.