Problem 13 - Calculus and its applications

Question

Differentiate. $$ y=\log _{8} x $$

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{x \ln 8} \).

Step 1: Convert Logarithm to Natural Logarithm

To differentiate the function with a base other than 'e', first convert the logarithm into natural logarithms using the change of base formula. For a logarithm with base 8, expressed as \(y = \log_8 x\), rewrite it as \(y = \frac{\ln x}{\ln 8}\). This conversion allows us to use derivatives we know.

Step 2: Differentiate Using the Derivative of Natural Logarithm

Differentiate the function \(y = \frac{\ln x}{\ln 8}\). Since \(\ln 8\) is a constant, the derivative of \(y\) with respect to \(x\) is \( \frac{1}{\ln 8} \cdot \frac{d}{dx}(\ln x) \). The derivative of \(\ln x\) is \(\frac{1}{x}\), so: \( \frac{d}{dx}(\ln x) = \frac{1}{x} \).

Step 3: Simplify the Expression

Substitute the derivative of \(\ln x\) back into the expression: \( \frac{d}{dx}y = \frac{1}{\ln 8} \cdot \frac{1}{x} = \frac{1}{x \ln 8} \). This is the expression for the derivative of \(y = \log_8 x\).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a method used to differentiate complicated functions by taking the natural logarithm of both sides of an equation. It is particularly useful when dealing with functions involving exponents and logarithms of arbitrary bases.

By converting to natural logarithms, we can utilize simpler derivative rules that are well-documented. The process involves:

• Taking the natural logarithm of both sides of an equation to simplify the expression.
• Applying the rules of logarithms, like the product, quotient, or power rule, to break down the expression.
• Finally, differentiating using the derivative rules of natural logarithms.

Logarithmic differentiation is a great tool when faced with differentiating functions with difficult bases that are not powers of the base "e." By simplifying the function first, you often convert a daunting derivative into a manageable calculation.

Natural Logarithm

A natural logarithm is a logarithm to the base "e," where "e" is approximately equal to 2.71828. This is denoted as \( \ln x \). Natural logarithms are essential in calculus because they simplify the process of differentiation.

In the context of differentiation, the derivative of \( \ln x \) is straightforward:

• The derivative of \( \ln x \) is \( \frac{1}{x} \), a simple and fundamental result in calculus.
• This differentiation is consistent, regardless of the value of "x," as long as "x" is positive.

The natural logarithm appears frequently in areas involving growth rates, decay, and many natural processes. Understanding how to differentiate natural logarithms lays a strong foundation for handling more complex logarithmic functions.

Change of Base Formula

The change of base formula is a mathematical tool that allows us to convert logarithms from one base to another. It is especially helpful when the base of a logarithm is not the standard base 10 or the natural base "e."

The formula states that for any base \( a \) and \( b \), the logarithm \( \log_a x \) can be expressed in terms of the natural logarithm (or any other base) using:

• \[ \log_a x = \frac{\ln x}{\ln a} \]

This transformation is critical in calculus when differentiating logarithmic functions not in base "e." By using the change of base formula, we can rewrite a difficult logarithm into a form amenable to natural logarithm differentiation rules. This practice simplifies the differentiation process, making it easier to apply efficient calculus techniques.