Problem 114 - Calculus and its applications

Question

Differentiate. $$ f(x)=\log _{7} x $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \log_{7} x \) is \( f'(x) = \frac{1}{x \ln 7} \).

Step 1: Identify the Function and its Components

We need to differentiate the function \( f(x) = \log_{7} x \). This is a logarithm with base 7. To differentiate, we'll use the change of base formula to convert it to a natural logarithm, which simplifies differentiation.

Step 2: Apply Change of Base Formula

The change of base formula states \( \log_{a} b = \frac{\ln b}{\ln a} \). Applying it here, \( \log_{7} x = \frac{\ln x}{\ln 7} \). This allows us to differentiate \( \ln x \) with respect to \( x \).

Step 3: Differentiate Using Natural Logarithm Rule

Differentiate \( f(x) = \frac{\ln x}{\ln 7} \). The derivative of \( \ln x \) is \( \frac{1}{x} \). Since \( \ln 7 \) is a constant, \( f'(x) = \frac{1}{\ln 7} \cdot \frac{1}{x} \).

Step 4: Write the Final Derivative

Combine the results from the previous step to give the final expression for the derivative: \( f'(x) = \frac{1}{x \ln 7} \). This is the derivative of the original function.

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is an elegant technique used when you need to differentiate a function that is expressed in terms of a logarithm. It allows us to simplify the differentiation process by utilizing the properties of logarithms. When we have a logarithm with an unfamiliar base, such as in the function \( f(x) = \log_{7} x \), logarithmic differentiation comes in handy. By applying certain formulas and rules, we can ease our way to the derivative.

• Convert the logarithm to a more familiar form that is easier to handle.
• This often involves converting to a natural logarithm with base \( e \).

Understanding and applying logarithmic differentiation properly is crucial for simplifying the differentiation of complex logarithmic functions. It prepares the function for the final differentiation step, making your calculus journey smoother.

Change of Base Formula

The change of base formula is a key tool in calculus to help us work with logarithms of any base. This formula allows us to convert a logarithmic function to a different base, typically the natural logarithm which is more convenient to differentiate.
This formula states: \( \log_{a} b = \frac{\ln b}{\ln a} \). In the problem \( f(x) = \log_{7} x \), the goal is to express this function using the natural logarithm. Applying the change of base formula, we rewrite:

• \( \log_{7} x = \frac{\ln x}{\ln 7} \)

This conversion simplifies the differentiation process since the natural logarithm has well-known differentiation rules. By changing the base, we can use the rule for differentiating natural logarithms effectively, making calculus operations more straightforward.

Natural Logarithm Differentiation

Differentiation of natural logarithms is far simpler than with other bases, thanks to the convenient derivative of \( \ln x \). When differentiating \( \ln x \), the derivative is simply \( \frac{1}{x} \). This simplicity and efficiency are why using natural logarithms in calculus is so prevalent.
Once we transform our original function \( \log_{7} x = \frac{\ln x}{\ln 7} \) using the change of base formula, differentiating it becomes direct:

• The derivative of \( \ln x \) is \( \frac{1}{x} \).
• Since \( \ln 7 \) is a constant, its differentiation effect is straightforward: divide the derivative of \( \ln x \) by this constant.

Thus, the differentiation process results in the derivative \( f'(x) = \frac{1}{x \ln 7} \). This result highlights the utility of natural logarithms in calculus, as they simplify expressions and make differentiation more manageable.