Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated derivative. \(D_{x} \log _{3} e^{x}\)

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{\ln(3)} \).

Step 1: Rewrite the logarithm as a natural logarithm

Recall that a logarithm with base 3 can be rewritten using the change of base formula: \( \log_{3}(e^x) = \frac{\ln(e^x)}{\ln(3)} \). This helps to simplify the differentiation process using natural logarithms.

Step 2: Simplify the expression

Further simplify the expression \( \frac{\ln(e^x)}{\ln(3)} \) by using the property of logarithms: \( \ln(e^x) = x \ln(e) = x \). Therefore, the expression becomes \( \frac{x}{\ln(3)} \).

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

Using basic differentiation rules, differentiate \( \frac{x}{\ln(3)} \) with respect to \(x\). The derivative of \( x \) is \( 1 \). Therefore, the derivative is \( \frac{1}{\ln(3)} \).

Step 4: State the final answer

The derivative \( D_{x} \log_{3}(e^x) \) is \( \frac{1}{\ln(3)} \). This simplifies from the differentiation of \( \frac{x}{\ln(3)} \).

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool that simplifies the process of working with logarithms of different bases. In math, the formula allows us to express a logarithm in terms of a different base, typically the natural logarithm. This is especially useful when you're calculating derivatives, like in our original exercise.
The change of base formula is expressed as:

• \( \log_b(x) = \frac{\ln(x)}{\ln(b)} \)

This means you can convert a logarithm of base \( b \) to a natural logarithm, which is very handy since natural logarithms are easier to differentiate. In our example, this formula was applied to convert \( \log_3(e^x) \) into \( \frac{\ln(e^x)}{\ln(3)} \).
This conversion simplifies the differentiation process, so you can use simpler rules associated with natural logs, making the entire procedure more straightforward.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm to the base of the mathematical constant \( e \), where \( e \approx 2.718 \). The natural logarithm is unique because it has special properties that make mathematical operations simpler, especially differentiation.
One key property of natural logarithms is \( \ln(e^x) = x \). This is because the exponential and natural logarithm functions are inverse functions.

• If you have \( \ln(e^x) \), you can simplify directly to \( x \) since \( \ln(e) \) equals \( 1 \).

In the exercise, this property was used to simplify \( \ln(e^x) \) to \( x \). Hence, our transformed expression \( \frac{\ln(e^x)}{\ln(3)} \) simplifies further to \( \frac{x}{\ln(3)} \).
This simplification showcases the beauty of natural logs in calculus, where experience with these properties can save you much time and effort.

Differentiation Rules

Differentiation is a fundamental concept in calculus used to find the rate at which a function is changing. Basic differentiation rules are essential when dealing with logarithmic functions, as they help compute derivatives effectively.
When you're finding the derivative of a straightforward function like \( x \), it results in a simple answer:\( \frac{d}{dx}(x) = 1 \).

• If a function is scaled by a constant, like \( \frac{x}{\ln(3)} \), the derivative involves the constant multiplied by the derivative of \( x \).

Thus, when differentiating \( \frac{x}{\ln(3)} \), the constant remains the same, and the derivative of \( x \) is \( 1 \). Therefore, the derivative becomes \( \frac{1}{\ln(3)} \).
These rules underline the simplicity of differentiating linear functions and serve as a reminder of how constants behave during differentiation. Knowing these basic rules allows you to tackle more complex functions with ease.