Problem 75 - Precalculus

Question

Graph the logarithmic function using transformation techniques. State the domain and range of \(f\). $$\ln (x+2)$$

Step-by-Step Solution

Answer

Domain is x > -2; range is all real numbers.

Step 1: Identify the Base Function

The function we are given is \( f(x) = \ln(x+2) \). The base function of a natural logarithm is \( g(x) = \ln(x) \). This function \( \ln(x) \) has a vertical asymptote at \( x = 0 \), a domain of \( x > 0 \), and a range of all real numbers.

Step 2: Determine the Transformation

The given function \( f(x) = \ln(x+2) \) involves a horizontal shift of the base function \( g(x) = \ln(x) \). Specifically, it is shifted left by 2 units. This transformation comes from the \((x+2)\) inside the logarithm.

Step 3: Apply the Horizontal Shift

When you shift \( \ln(x) \) left by 2 units, every point on the curve moves 2 units to the left. The vertical asymptote, originally at \( x = 0 \), is now at \( x = -2 \). Thus, the function \( f(x) \) will be undefined for \( x \leq -2 \).

Step 4: Determine the Domain and Range

After applying the transformation, the domain of \( f(x) = \ln(x+2) \) becomes \( x > -2 \). The range of a logarithmic function, including this transformed one, remains all real numbers \(( -\infty, \infty )\).

Key Concepts

Domain and Range of Logarithmic Functions

When graphing logarithmic functions, understanding domain and range is key. The domain refers to all possible input values for the function. For the base natural logarithm function, \( \ln(x) \), the domain is \( x > 0 \). This is because you can't take the logarithm of zero or negative numbers in real numbers. When transformations are applied, like with \( \ln(x+2) \), the domain shifts accordingly. Here, because of the horizontal shift left by 2 units, the new domain becomes \( x > -2 \). This means you must input values greater than \(-2\) into the function.The range, on the other hand, represents all possible output values. For a natural logarithmic function and its transformed forms, the range remains as \( (-\infty, \infty) \). This suggests that as \( x \) increases, the function's output can cover any real number.

• Key Insight: Logarithmic functions are always defined for positive numbers. Adjustments shift the domain accordingly.
• Remember: The range spans all real numbers, but the domain depends on transformations.

Understanding Logarithmic Transformations

Transformations in functions involve shifts, reflections, stretches, and compressions. In the context of logarithmic transformations, we often deal with shifts. For \( \ln(x+2) \), the function involves a horizontal shift. Adding 2 inside the logarithm moves the function's graph 2 units to the left, resulting in \( \ln(x+2) \). It means every point on the original \( \ln(x) \) curve has shifted leftward by two units.A key result of this transformation is the location of the vertical asymptote, which moves from \( x = 0 \) for \( \ln(x) \) to \( x = -2 \) for \( \ln(x+2) \). This shift does not affect the range but redefines the start of the domain.

• Shifting Left: Adds a value inside the parentheses \( (x+c) \).
• New Asymptote: Occurs where \( (x+2)=0 \) or \( x=-2 \).
• Visual Changes: Plots of functions slide along the x-axis according to the transformation applied.

Insight into Natural Logarithms

Natural logarithms are denoted as \( \ln \), and they use Euler's number \( e \) as their base. Euler’s number, approximately 2.718, is a fundamental constant in mathematics, especially in calculus. The natural logarithm function is the inverse of the exponential function \( e^x \). This implies that if \( y = \ln(x) \), then \( e^y = x \).These logarithms play a crucial role in various applications, like calculating compound interest and population growth. Their graphs provide a gradual curve depicting slow growth; this is why the logarithmic function graphs have a steep slope initially that becomes more moderate.

• Base \( e \): Critical constant approximately equal to 2.718.
• Inverse Function: Reverse of exponential \( e^x \).
• Applications: Used in exponential decay and growth models, calculus, and financial calculations.