Problem 77 - Thomas Calculus in SI Units

Question

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{10} e^{x}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{\ln(10)} \).

Step 1: Apply the change of base formula for the logarithm

The expression is given as \( y = \log_{10}(e^x) \). Using the change of base formula, \( \log_{a}(b) = \frac{\ln(b)}{\ln(a)} \), rewrite \( y \) as \( y = \frac{\ln(e^x)}{\ln(10)} \).

Step 2: Simplify the logarithmic expression

Simplify \( \ln(e^x) \). Since the natural logarithm and the exponential are inverse functions, \( \ln(e^x) = x \). Thus, \( y = \frac{x}{\ln(10)} \).

Step 3: Differentiate the simplified expression with respect to \( x \)

The expression \( y = \frac{x}{\ln(10)} \) is in the form of a constant multiplied by \( x \). Differentiate it with respect to \( x \):\[ \frac{dy}{dx} = \frac{1}{\ln(10)} \cdot (1) = \frac{1}{\ln(10)} \].

Key Concepts

Change of Base Formula

The change of base formula is an important tool in calculus and logarithms. It allows us to switch from one base to another, specifically to the more convenient base of natural logarithms, denoted as \( \ln \).
This can be particularly helpful when performing operations involving derivatives or integrals.The formula is expressed as follows:

• \( \log_{a}(b) = \frac{\ln(b)}{\ln(a)} \)

In our exercise, we started with the expression \( y = \log_{10}(e^x) \).
By applying the change of base formula, we rewrite this as \( y = \frac{\ln(e^x)}{\ln(10)} \). Switching to the natural logarithm base simplifies many calculus tasks, such as differentiation, as it seamlessly integrates with known derivative rules.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. Natural logarithms have unique properties useful in calculus:

• \( \ln(e) = 1 \)
• \( \ln(e^x) = x \)
• Derivative: \( \frac{d}{dx}\ln(x) = \frac{1}{x} \)

In our exercise:

• We applied \( \ln(e^x) = x \) to the expression \( \ln(e^x) \), simplifying the task.

Using natural logarithms can turn complex expressions into simpler forms, making derivatives easier to compute and understand.

Exponential Function

Exponential functions are pivotal in calculus. They take the form \( b^x \), where \( b > 0 \) and \( x \) is any real number.The natural exponential function with base \( e \), written as \( e^x \), is especially noteworthy.

• It is the inverse of the natural logarithm.
• The derivative of \( e^x \) is uniquely simple and remains the same: \( \frac{d}{dx}e^x = e^x \).

In our problem, simplifying the expression \( \ln(e^x) = x \) used the inverse property effectively.Recognizing these relationships helps to easily derive results, simplifying calculus operations.
Consequently, understanding the interplay between \( e^x \) and \( \ln(x) \) is fundamental for efficiently tackling various calculus problems.