Question

Find the domain of each function. $$ f(x)=\sqrt{\ln x} $$

Short answer

The domain of \(\text{f(x) = } \sqrt{\text{ln}(x)}\) is \(\text{[1, }\text{\textinfty )}\).

Step-by-step solution

Understand the Inner Function

The function inside the square root is \(\text{ln}(x)\). The natural logarithm \(\text{ln}(x)\) is only defined for \(\text{x > 0}\).

Consider the Square Root Function

For \(\text{f(x) = } \sqrt{\text{ln}(x)}\), the expression inside the square root, which is \(\text{ln}(x)\), must be greater than or equal to zero because the square root function is only defined for non-negative numbers.

Set Up the Inequality

To find the domain, solve the inequality \(\text{ln}(x) \geq 0\).

Solve the Inequality

The inequality \(\text{ln}(x)\) \geq 0\ is equivalent to \(\text{x} \geq \text{e}^0\) because the natural logarithm of 1 is 0. Hence, \(\text{x} \geq 1\).

Determine the Domain

Since \(\text{x}\) must be greater than or equal to 1, the domain of the function \(\text{f(x) = } \sqrt{\text{ln}(x)}\) is \(\text{[1, }\text{\textinfty )}\).

Key concepts

Natural Logarithm

The natural logarithm, often written as \(\text{ln}(x)\), is a mathematical function that is only defined for positive numbers. This means if you see \(\text{ln}(x)\), the value inside the parentheses, \(\text{x}\), must be greater than zero. The natural logarithm of a number gives the power to which you must raise \(\text{e}\) (approximately 2.71828) to get that number. For example, \(\text{ln}(1) = 0\) because \(\text{e}^0 = 1\). Similarly, \(\text{ln}(e) = 1\) because \(\text{e}^1 = e\).
The natural logarithm grows slowly compared to other common functions, meaning that for even large values of \(\text{x}\), \(\text{ln}(x)\) does not increase very rapidly. Understanding the domain limitations of the natural logarithm is crucial when it's embedded in other functions, such as square root functions.

Square Root Function

A square root function is typically written as \(\text{f(x) = } \sqrt{x}\). For this function to be real and valid, the value inside the square root (called the radicand) must be greater than or equal to zero. This is because the square root of a negative number is not a real number (in the realm of standard real number arithmetic).
Applying this to our specific exercise, \(\text{f(x) = } \sqrt{\text{ln}(x)}\), we need \(\text{ln}(x)\) to be at least zero for \(\text{f(x)}\) to be defined. If \(\text{ln}(x) < 0\), it would result in an undefined or non-real number, which we want to avoid.

Inequalities in Algebra

Inequalities are used to compare values and express that one value is larger, smaller, or equal to another. Solving inequalities involves finding the range of values that satisfy the inequality.
For instance, in our exercise, we need to solve \(\text{ln}(x) \geq 0\). This inequality can be understood by knowing that \(\text{ln}(x)\) is zero when \(\text{x} = 1\), and it is positive for all \(\text{x} > 1\). Therefore, the inequality holds true when \(\text{x} \geq 1\).
When solving such inequalities, it’s important to remember the properties of logarithmic and exponential functions, such as \(\text{ln}(1) = 0\) and \(\text{ln}(e) = 1\), to determine the correct domain for the function. Inequalities often are accompanied by conditions that must be fulfilled for the result to be valid, like in our case where \(\text{x}\) needs to be greater than or equal to 1.