Problem 42 - Essential Calculus Early Transcendentals

Question

(a) What is the natural logarithm? (b) What is the common logarithm? (c) Sketch the graphs of the natural logarithm function and the natural exponential function with a common set of axes.

Step-by-Step Solution

Answer

(a) The natural logarithm has base \(e\); (b) the common logarithm has base 10; (c) sketch respective curves on the same axes.

Step 1: Definition of Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. The natural logarithm of a number \(x\) is the exponent to which \(e\) must be raised to obtain \(x\). For example, \(\ln(e) = 1\).

Step 2: Definition of Common Logarithm

The common logarithm, denoted as \(\log_{10}(x)\), is the logarithm to the base 10. It is the exponent to which 10 must be raised to yield a given number \(x\). For example, \(\log_{10}(10) = 1\).

Step 3: Sketching the Graphs

For sketching the graphs of the natural logarithm function \(y = \ln(x)\) and the natural exponential function \(y = e^x\), create a coordinate system with the x-axis representing the input values and the y-axis representing the output values. The graph of \(y = \ln(x)\) will curve upwards and pass through the point (1,0), as the natural logarithm of 1 is 0. The graph of \(y = e^x\) will rise exponentially and pass through the point (0,1) because \(e^0 = 1\). Draw both graphs to demonstrate their characteristic shapes on the same axes.

Key Concepts

Natural Logarithm

The natural logarithm is an essential concept in mathematics, noted by the symbol \( \ln(x) \). It specifically refers to the logarithm to the base \( e \), a notable constant roughly equal to 2.71828. Understanding \( \ln(x) \) involves recognizing it as the power to which \( e \) must be raised to produce the number \( x \). For example, \( \ln(e) = 1 \) since raising \( e \) to the power of 1 gives \( e \). In practice:

• \( \ln(x) \) is widely used in calculus, notably when solving equations involving exponential growth and decay.
• The natural logarithm is integral to scientific computations, including those involving continuous growth processes such as population growth or radioactive decay.

Hence, it's integral in fields ranging from sciences to engineering and statistics, demonstrating its versatile utility.

Common Logarithm

The common logarithm is another significant type of logarithm, symbolized as \( \log_{10}(x) \). This form of logarithm uses 10 as its base, making it highly applicable in various scientific and engineering contexts where base 10 is natural, such as dealing with orders of magnitude. A few key aspects include:

• The common logarithm answers the question: "To what power must 10 be raised to result in \( x \)?"
• For instance, with \( \log_{10}(1000) = 3 \), 10 must be raised to the power of 3 to produce 1000.
• It's used in a multitude of applications, including measuring sound intensity (decibels) and the Richter scale for earthquakes.

Its practical applications across various fields underscore the importance of understanding and using common logarithms effectively.

Graphing Functions

Visualizing functions like logarithmic and exponential functions can significantly aid in comprehending their behavior.When graphing the natural logarithm \( y = \ln(x) \):

• Begin by establishing a coordinate system where the x-axis represents input values and the y-axis indicates output values.
• The curve of \( \ln(x) \) rises gently, passing through the point (1,0), since \( \ln(1) = 0 \).
• As \( x \) increases, the logarithm continues to rise slowly without bound.

Graphing the natural exponential function \( y = e^x \) on the same set of axes shows:

• The curve passes through the point (0,1) as \( e^0 = 1 \).
• It exemplifies exponential growth, rapidly increasing as \( x \) becomes greater.

Juxtaposing these graphs can illuminate the inverse relationship between the exponential function and the natural logarithm, enhancing the understanding of how they complement each other in mathematical contexts.