Questions: What is the domain of g(x)=ln(5x-13)?

What is the domain of g(x)=ln(5x-13)?
Transcript text: What is the domain of $g(x)=\ln (5 x-13)$ ?
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Solution

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Solution Steps

To find the domain of the function \( g(x) = \ln(5x - 13) \), we need to determine the values of \( x \) for which the argument of the natural logarithm (i.e., \( 5x - 13 \)) is greater than 0. This is because the natural logarithm is only defined for positive arguments.

Step 1: Identify the Argument of the Logarithm

The function \( g(x) = \ln(5x - 13) \) requires that the argument \( 5x - 13 \) be greater than 0 for the logarithm to be defined. Therefore, we set up the inequality: \[ 5x - 13 > 0 \]

Step 2: Solve the Inequality

To solve the inequality, we isolate \( x \): \[ 5x > 13 \] \[ x > \frac{13}{5} \]

Step 3: Express the Domain

The solution indicates that \( x \) must be greater than \( \frac{13}{5} \). Thus, the domain of the function \( g(x) \) can be expressed in interval notation as: \[ \left( \frac{13}{5}, \infty \right) \]

Final Answer

The domain of \( g(x) = \ln(5x - 13) \) is \(\boxed{\left( \frac{13}{5}, \infty \right)}\).

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