Problem 81 - College Algebra Essentials

Question

Use a graphing utility and the change-of-base property to graph each function. \(y=\log _{2}(x+2)\)

Step-by-Step Solution

Answer

The graph of the function \(y=\log _{2}(x+2)\) is represented by the curve obtained after entering the transformed function \(y=\frac{\ln (x+2)}{\ln 2}\) into a graphing utility.

Step 1: Understanding the Change-of-Base Property

The change-of-base property of logarithms states that for any positive base \(a\), \(b\), (\(a\neq1\), \(b\neq1\)) and any real number \(x\), the function \(\log _{b}(a)\) can be expressed in terms of natural logarithms (base e) as \(\log _{b}(a) = \frac{\ln a}{\ln b}\)

Step 2: Applying the Change-of-Base Property

Apply this property to the given function \(y=\log_2(x+2)\). It can be expressed in terms of natural logarithms (base e) as \(y=\frac{\ln (x+2)}{\ln 2}\). This transformation allows the graphing utility, usually set to base e, to understand and correctly plot the function.

Step 3: Graphing the Function

Input the transformed function \(y=\frac{\ln (x+2)}{\ln 2}\) into the graphing utility. The graph will represent the function \(y=\log_2(x+2)\).

Key Concepts

Graphing Logarithmic Functions

When graphing logarithmic functions, it's essential to understand their basic properties and transformations. A logarithmic function typically has the form:

• \( y = \log_b(x) \)

where \( b \) is the base of the logarithm. This base determines the steepness and direction of the graph.
For example, with \( y = \log_2(x+2) \), the graph will shift horizontally. This happens because of the "\(+2\)" inside the parentheses, translating the graph two units to the left.
Inverting the logarithmic function involves converting it into an exponential function, reflecting the graph across the line \( y = x \). The unique feature of log functions, irrespective of the base, is their vertical asymptote at \( x=0 \) before any horizontal shifts.
Fortunately, modern graphing utilities make plotting easy. By converting the function using the change-of-base property, these tools can graph functions with various bases, not just natural or common logarithms.

Logarithmic Transformations

Applying transformations to logarithmic functions is a vital skill that can help to shift, stretch, or compress a graph. Transformations can include:

• Horizontal Shifts: Changing the function to \( y = \log_b(x+h) \) shifts the graph horizontally by \( h \) units. A positive \( h \) value moves the graph left, while a negative \( h \) value shifts it right.
• Vertical Shifts: Adding a constant \( k \) results in \( y = \log_b(x) + k \), moving the graph up or down by \( k \) units.
• Stretching/Compressing: Multiplying the logarithmic function by a constant \( a \) changes its steepness. If \( y = a\log_b(x) \), the graph stretches vertically if \( a > 1 \) and compresses if \( 0 < a < 1 \).

These transformations can be combined in various ways to obtain the desired graph. In the function \( y = \log_2(x+2) \), and particularly when re-written using the change-of-base property as \( y = \frac{\ln(x+2)}{\ln 2} \), a horizontal shift left by two units is clearly seen. Understanding how to manipulate these transformations allows for a greater grasp of graph behavior and plot accuracy.

Natural Logarithms

Natural logarithms use the base \( e \), an irrational number approximately equal to 2.71828. They arise naturally in mathematical calculations, especially in continuous growth phenomena. The notation for a natural logarithm is \( \ln(x) \), and it's a common base used in calculus and sciences because of its nice properties in integration and differentiation.
Natural logarithms serve a special purpose in the change-of-base formula. The formula \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \) allows converting any logarithm with a base different from \( e \) into one that uses natural logarithms. This capability is crucial when using graphing calculators or software, as most are configured for natural logs. By using the relation to natural logs, we transformed the function \( y = \log_2(x+2) \) into \( y = \frac{\ln(x+2)}{\ln 2} \), which was easily plotted using graphing utilities. Thus, understanding natural logarithms and the change-of-base property is essential for accurate function representation and problem-solving in math-related fields.