Problem 20 - Calculus for Biology and Medicine

Question

In Problems \(15-24\), find the values of \(x \in \mathbf{R}\) for which the given functions are continuous. $$ f(x)=\ln (x-2) $$

Step-by-Step Solution

Answer

The function is continuous for \( x > 2 \).

Step 1: Understand the Domain of Natural Logarithm

The natural logarithm function \( \ln(x) \) is defined only for positive values of \( x \). Therefore, \( x-2 \) must be greater than zero for \( \ln(x-2) \) to be defined.

Step 2: Set the Inequality

To find when \( \ln(x-2) \) is defined, set the inequality \( x-2 > 0 \).

Step 3: Solve the Inequality

Solve the inequality from Step 2: \( x-2 > 0 \). Add 2 to both sides to find \( x > 2 \).

Step 4: Conclusion

The function \( f(x) = \ln(x-2) \) is continuous for \( x > 2 \) because the natural logarithm is continuous for all positive values.

Key Concepts

Natural Logarithm

The natural logarithm function, represented as \( \ln(x) \), holds a special place in mathematics due to its elegant properties and frequent occurrence in calculus and exponential growth models. The natural logarithm is defined only for positive numbers, meaning \( x \) must be greater than zero for \( \ln(x) \) to exist. This concept stems from the logarithm's relationship with exponentiation, where it represents the power to which the base \( e \) (approximately 2.71828) must be raised to produce the number \( x \). Understanding this requirement is crucial because it influences the domain of logarithmic functions. With the context of continuity in calculus, we need to ensure that the argument of the logarithm (here, \( x-2 \)) remains positive. Only then can the function be defined continuously for all valid \( x \) values.

Domain of a Function

The domain of a function is essentially all the possible inputs (or \( x \) values) for which the function produces valid outputs. In the context of the function \( f(x) = \ln(x-2) \), our focus is on determining when the expression \( x-2 \) is greater than zero. This arises because the argument of a natural logarithm must be positive.Steps to Determine the Domain:

• Identify the expression within the logarithm: \( x-2 \).
• Set up the inequality that this expression must satisfy: \( x-2 > 0 \).
• Solve this inequality to find the permissible range for \( x \).

By adding 2 to both sides of the inequality, we conclude that \( x > 2 \). Hence, the domain of \( f(x) \) is all real numbers greater than 2. This ensures that the function is continuous and well-defined.

Inequalities

Inequalities allow us to express a range of values rather than a single point, which is particularly useful for defining domains in functions like \( \ln(x-2) \). In the solution to our problem, we dealt with the inequality \( x-2 > 0 \) to determine when the natural logarithm is defined.Understanding and Solving Inequalities:

• Recognize the inequality: Here, \( x-2 > 0 \).
• Manipulate the inequality to isolate \( x \): Add 2 to both sides to find \( x > 2 \).
• Interpret the solution: \( x > 2 \) means any real number greater than 2 can be substituted into the function without causing undefined behavior.

Inequalities are foundational in determining where functions are valid, ensuring continuity, and identifying the regions where the function performs smoothly without breaks or jumps. In this example, solving the inequality provided us with the domain where \( f(x) \) is continuous, reinforcing how inequalities directly impact function analysis.