Problem 38 - College Algebra

Question

For the following exercises, sketch the graphs of each pair of functions on the same axis.$$f(x)=\log _{4}(x) \text { and } g(x)=\ln (x)$$

Step-by-Step Solution

Answer

\( f(x) = \log_4(x) \) is slower increasing, passes through (4,1); \( g(x) = \ln(x) \) is faster, passes through (e,1). Both are undefined at \( x \leq 0 \).

Step 1: Understanding the functions

We have two functions: \( f(x) = \log_4(x) \) and \( g(x) = \ln(x) \). Here, \( \log_4(x) \) is the logarithm of \( x \) with base 4, and \( \ln(x) \) is the natural logarithm of \( x \) with base \( e \). Both functions are defined for \( x > 0 \).

Step 2: Finding key points for \( f(x) = \log_4(x) \)

For \( f(x) \), find some key points to help in sketching the graph.\- \( f(1) = \log_4(1) = 0 \)- \( f(4) = \log_4(4) = 1 \)- \( f(16) = \log_4(16) = 2 \)These points represent integer values where the function crosses grid lines on the graph.

Step 3: Finding key points for \( g(x) = \ln(x) \)

For \( g(x) \), also identify key points:- \( g(1) = \ln(1) = 0 \)- \( g(e) = \ln(e) = 1 \)- \( g(e^2) = \ln(e^2) = 2 \)These will help us plot the natural logarithm effectively.

Step 4: Identifying behavior of \( f(x) = \log_4(x) \) and \( g(x) = \ln(x) \)

Both functions are undefined for \( x \leq 0 \) and intercept the y-axis at \( x = 1 \) each at \( f(1) = 0 \) and \( g(1) = 0 \). Both are increasing as \( x \) increases, but \( f(x) = \log_4(x) \) grows more slowly compared to \( g(x) = \ln(x) \) as base 4 is greater than \( e \).

Step 5: Sketching the graph

Graph both functions on the same axis, marking key points.1. Plot \( f(x) = \log_4(x) \) through points \( (1, 0), (4, 1), (16, 2) \).2. Plot \( g(x) = \ln(x) \) through points \( (1, 0), (e, 1), (e^2, 2) \).3. Both graphs should approach the y-axis asymptotically, never touching it, and should continue to increase. Note \( g(x) \) rises faster because base \( e \) is smaller than 4.

Key Concepts

Logarithmic Functions

Logarithmic functions are essential mathematical functions used frequently in various scientific fields. They are the inverse of exponential functions and help us solve equations involving exponents. In the expression \( y = \log_b(x) \), \( b \) is the base and \( x \) is the argument. The expression simply tells us what power, \( y \), is necessary for base \( b \) to equal \( x \).
Logarithms with different bases have unique behaviors, which significantly affect their graph's shape. Nevertheless, all logarithmic functions demonstrate similar general properties:

• They are defined only for positive \( x \), meaning \( x > 0 \), representing situations where logarithms are real numbers.
• The y-intercept occurs at \((1, 0)\) since any number raised to the power zero is 1.
• They feature a vertical asymptote along the y-axis (x=0), which the functions approach but never touch.
• The graphs are always increasing, showing a positive relationship between \( y \) and \( x \).

Understanding these properties is crucial for sketching the graphs accurately.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a specific logarithm with base \( e \), where \( e \approx 2.718 \). This constant, \( e \), is an irrational number with unique properties making the natural logarithm extensively useful in calculus, compounding interest problems, and natural growth scenarios.
When working with natural logarithms, consider these key aspects:

• \( \ln(1) = 0 \) because any number to the power of zero is 1.
• \( \ln(e) = 1 \) since \( e^1 = e \).
• \( \ln(e^2) = 2 \) as the expression directly reflects the inverse property of exponentiation.

Natural logarithms provide smooth transitions over data and are often preferred in scientific computations due to their unique growth rate properties linked to continuous compounding scenarios.

Function Sketching

Sketching the graph of a function involves plotting points and understanding the general shape and direction of the graph. The task is easier once the key properties of a function, as outlined before, are known. For log functions such as \( f(x) = \log_4(x) \) and \( g(x) = \ln(x) \), sketching involves identifying specific points and behaviors:

• Start by marking the y-intercept, which traditionally occurs at \( (1, 0) \).
• Plot additional points like \( (4, 1) \) for \( \log_4(x) \) and \( (e, 1) \) for \( \ln(x) \), highlighting the increment process.
• Reflect on the growth rate: \( \ln(x) \) increases faster than \( \log_4(x) \), leading to a steeper curve.
• Remember, both functions approach the y-axis asymptotically, demonstrating steadily rising behavior over time.

Combining these principles, sketch clean and representative graphs, respecting the axis, intercepts, and general trends.