Question

State the domain of \(f(x)=\ln (5 x+2)\)

Short answer

The domain of \(f(x) = \text{ln}(5x + 2)\) is \((- \frac{2}{5}, \infty)\).

Step-by-step solution

Understand the Logarithmic Function

The natural logarithm function, denoted as \(\text{ln}(x)\), is defined only for positive arguments, that is, \(\text{ln}(x)\) is defined if and only if \(x > 0\). Thus, any function inside the natural logarithm must be greater than zero.

Set the Argument Greater Than Zero

For the function \(f(x) = \text{ln}(5x + 2)\), the argument (5x + 2) must be greater than zero. So, set up the inequality: \(5x + 2 > 0\).

Solve the Inequality

Solve the inequality \(5x + 2 > 0\) to find the domain of the function.\(5x + 2 > 0\) Subtract 2 from both sides:\(5x > -2\). Divide both sides by 5:\(x > -\frac{2}{5}\).

State the Domain

The domain of \(f(x) = \text{ln}(5x + 2)\) is the set of all x values such that \(x > -\frac{2}{5}\). In interval notation, this is written as \((- \frac{2}{5}, \infty)\).

Key concepts

natural logarithm

The natural logarithm, represented as \(\text{ln}(x)\), is a special logarithm with the base e (approximately 2.718).
It is important to understand that natural logarithm functions are only defined for positive values of x.
This means \(\text{ln}(x)\) only exists when x > 0, making it crucial to ensure any arguments within the natural logarithm are positive.
For instance, if we have \(\text{ln}(5x + 2)\), the expression inside, 5x + 2, must be greater than 0.

inequalities

Inequalities are mathematical expressions involving symbols like >, <, ≥, and ≤ to show the relative sizes of two values.
To find the domain of \(\text{ln}(5x + 2)\), we need to set up an inequality because the argument of the logarithm must be positive.
We set the inequality as 5x + 2 > 0.
Solving this step-by-step involves:

• Subtracting 2 from both sides: 5x + 2 - 2 > 0 - 2 gives 5x > -2
• Dividing both sides by 5: \(\frac{5x}{5} > \frac{-2}{5}\), which simplifies to x > -\frac{2}{5}

This tells us that x must be greater than -2/5 to ensure the argument of the logarithm is positive.

interval notation

Interval notation is a way to represent the set of all numbers between two endpoints.
It uses parentheses ( ) for open intervals, which do not include the endpoints, and brackets [ ] for closed intervals, which do include the endpoints.
In our case, the solution to the inequality x > -2/5 is an open interval because -2/5 is not included.
Therefore, the domain in interval notation is written as \((- \frac{2}{5}, \infty)\).
The left parenthesis indicates that -2/5 is not included, while the infinity symbol \(\infty\) suggests that there is no upper limit.

function domain

The domain of a function includes all the input values (x-values) that will produce a valid output.
For the logarithmic function \(f(x) = \text{ln}(5x + 2)\), the domain is determined by ensuring that the argument of \(\text{ln}\) remains positive.
This involves setting up and solving the inequality 5x + 2 > 0, which gives x > -2/5.
Therefore, the domain of \(\text{ln}(5x + 2)\) is all x such that x > -2/5.
In interval notation, we represent this as \((- \frac{2}{5}, \infty)\), indicating the range of x-values for which the function is defined.