Given $y=\log _3(x+4)$, What Is The Domain?A. $x \ \textgreater \ -4$ B. \$x \ \textless \ -4$[/tex\] C. All Real Numbers

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Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. When dealing with logarithmic functions, it's essential to consider the properties of logarithms to determine the domain. In this article, we will explore the domain of the given logarithmic function $y=\log _3(x+4)$.

The Properties of Logarithms

A logarithmic function is defined as the inverse of an exponential function. The general form of a logarithmic function is $y=\log _a(x)$, where aa is the base of the logarithm. For the given function $y=\log _3(x+4)$, the base is 3.

One of the fundamental properties of logarithms is that the argument (the value inside the logarithm) must be positive. In other words, the expression x+4x+4 must be greater than 0 for the function to be defined.

Determining the Domain

To determine the domain of the given function, we need to find the values of xx for which the argument x+4x+4 is positive. This can be expressed as:

x+4>0x+4 > 0

Subtracting 4 from both sides of the inequality, we get:

x>−4x > -4

Therefore, the domain of the function $y=\log _3(x+4)$ is all real numbers greater than -4.

Conclusion

In conclusion, the domain of the given logarithmic function $y=\log _3(x+4)$ is all real numbers greater than -4. This can be expressed as:

x>−4x > -4

This means that any value of xx greater than -4 will result in a defined value of yy.

Answer

The correct answer is:

A. $x \ \textgreater \ -4$

Final Thoughts

In this article, we explored the domain of a logarithmic function $y=\log _3(x+4)$. We used the properties of logarithms to determine the values of xx for which the function is defined. The domain of the function is all real numbers greater than -4. This is an essential concept in mathematics, and understanding the domain of a function is crucial in solving problems involving logarithmic functions.

Additional Resources

For further reading on logarithmic functions and their properties, we recommend the following resources:

  • Khan Academy: Logarithmic Functions
  • Math Is Fun: Logarithmic Functions
  • Wolfram MathWorld: Logarithmic Function

These resources provide a comprehensive overview of logarithmic functions and their properties, including the domain of a logarithmic function.