The argument of the logarithm must be positive

Oct 13, 2025 by miodeya.com 61 views

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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 working with logarithmic functions, it's essential to consider the restrictions on the domain caused by the presence of the logarithm. In this article, we will explore how to find the domain of the function $$y=\log (1-4x)$$.

Understanding Logarithmic Functions

A logarithmic function is defined as the inverse of an exponential function. The general form of a logarithmic function is $$y=\log_b(x)$$, where $b$ is the base of the logarithm. The logarithmic function is defined only for positive real numbers, and the domain of the function is all real numbers greater than zero.

Restrictions on the Domain of Logarithmic Functions

When working with logarithmic functions, there are two main restrictions on the domain:

The argument of the logarithm must be positive: The expression inside the logarithm must be greater than zero. This is because the logarithm of a non-positive number is undefined.
The base of the logarithm must be positive: The base of the logarithm must be greater than zero. This is because the logarithm of a number with a non-positive base is undefined.

Finding the Domain of the Function $$y=\log (1-4x)$$

To find the domain of the function $$y=\log (1-4x)$$, we need to consider the restrictions on the domain caused by the presence of the logarithm.

Step 1: Identify the Argument of the Logarithm

The argument of the logarithm is the expression inside the logarithm, which is $1-4x$ .

Step 2: Determine the Restrictions on the Domain

To find the domain of the function, we need to determine the values of $x$ for which the argument of the logarithm is positive.

Step 3: Solve the Inequality

We need to solve the inequality $1-4x>0$ to find the values of $x$ for which the argument of the logarithm is positive.

1 - 4x > 0
-4x > -1
x < 1/4

Step 4: Write the Domain in Interval Notation

The domain of the function is all real numbers less than $1/4$ . We can write this in interval notation as $(-\infty, 1/4)$ .

Conclusion

In conclusion, the domain of the function $$y=\log (1-4x)$$ is all real numbers less than $1/4$ , which can be written in interval notation as $(-\infty, 1/4)$ .

Final Answer

The final answer is $\boxed{(-\infty, 1/4)}$ .

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Introduction

In our previous article, we explored how to find the domain of the function $$y=\log (1-4x)$$. In this article, we will answer some frequently asked questions related to finding the domain of logarithmic functions.

Q&A

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all real numbers greater than zero.

Q: What are the restrictions on the domain of a logarithmic function?

A: There are two main restrictions on the domain of a logarithmic function:

The argument of the logarithm must be positive: The expression inside the logarithm must be greater than zero.
The base of the logarithm must be positive: The base of the logarithm must be greater than zero.

Q: How do I find the domain of a logarithmic function?

A: To find the domain of a logarithmic function, you need to consider the restrictions on the domain caused by the presence of the logarithm. You need to identify the argument of the logarithm and determine the values of the variable for which the argument is positive.

Q: What is the domain of the function $$y=\log (x+1)$$?

A: To find the domain of the function $$y=\log (x+1)$$, we need to consider the restrictions on the domain caused by the presence of the logarithm. The argument of the logarithm is $x+1$ , which must be greater than zero. Therefore, the domain of the function is all real numbers greater than $-1$ , which can be written in interval notation as $(-1, \infty)$ .

Q: What is the domain of the function $$y=\log (x-2)$$?

A: To find the domain of the function $$y=\log (x-2)$$, we need to consider the restrictions on the domain caused by the presence of the logarithm. The argument of the logarithm is $x-2$ , which must be greater than zero. Therefore, the domain of the function is all real numbers greater than $2$ , which can be written in interval notation as $(2, \infty)$ .

Q: How do I write the domain of a logarithmic function in interval notation?

A: To write the domain of a logarithmic function in interval notation, you need to use the following notation:

$(-\infty, a)$ : This represents all real numbers less than $a$ .
$(a, \infty)$ : This represents all real numbers greater than $a$ .
** $(-\infty, a]$ $: This represents all real numbers less than or equal to $a$ .
$[a, \infty)$ : This represents all real numbers greater than or equal to $a$ .

Conclusion

In conclusion, finding the domain of a logarithmic function involves considering the restrictions on the domain caused by the presence of the logarithm. By identifying the argument of the logarithm and determining the values of the variable for which the argument is positive, you can find the domain of the function.

Final Answer

The final answer is $\boxed{(-\infty, 1/4)}$ .