Problem 14 - Calculus and its applications

Question

Differentiate. $$ y=\log _{4} x $$

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{x\ln 4} \).

Step 1: Convert to Natural Logarithm

The logarithm with base 4 can be converted to a natural logarithm (base e) using the change of base formula. This states \( \log_{b} a = \frac{\ln a}{\ln b} \). Apply this to \( y=\log_{4}x \), resulting in \[ y=\frac{\ln x}{\ln 4}. \]

Step 2: Differentiate Using the Chain Rule

Differentiate \( y = \frac{\ln x}{\ln 4} \). Use the constant multiple differentiation rule, which tells us we can differentiate separately: \[ \frac{dy}{dx} = \frac{1}{\ln 4} \cdot \frac{d}{dx}(\ln x). \] Since \( \frac{d}{dx}(\ln x) = \frac{1}{x} \), it follows that \[ \frac{dy}{dx} = \frac{1}{\ln 4} \cdot \frac{1}{x}. \]

Step 3: Simplify the Derivative

Now, simplify the expression \[ \frac{dy}{dx} = \frac{1}{x \ln 4}. \] This is the derivative of the function \( y = \log_{4}x \).

Key Concepts

Natural Logarithm

The natural logarithm is a logarithmic function with the base "e", where "e" is an irrational number approximately equal to 2.71828. It's often denoted as "ln". Natural logarithms are preferred in calculus due to their unique mathematical properties that simplify differentiation and integration. Here’s why they are essential:

• The derivative of the natural logarithm, \( \ln x \), is simply \( \frac{1}{x} \), making it easier to work with.
• Natural logarithms arise naturally in many areas of math, especially when dealing with exponential growth and decay.

A key point is that any logarithm with a different base can be converted to a natural logarithm using the change of base formula, allowing us to take advantage of their simpler derivatives.

Change of Base Formula

The change of base formula is a mathematical tool that allows us to convert a logarithm of one base into a logarithm of another base. This is particularly useful in calculus for functions involving logarithms. The formula is expressed as:

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

In the original exercise, we used this formula to convert a logarithm with base 4 into a natural logarithm (base e). By doing so, \( y = \log_{4} x \) transforms into \( y = \frac{\ln x}{\ln 4} \). This conversion is key to easily differentiating the function, leveraging the properties of natural logarithms. The base \( \ln 4 \) remains constant and only the natural logarithm of \( x \) demands differentiation.

Chain Rule

The chain rule is a fundamental concept in calculus used for differentiating composite functions. When you have a function within another function, the chain rule helps you differentiate effectively. Here's a simple way to understand it:

• If you have a function \( y = f(g(x)) \), the derivative \( \frac{dy}{dx} \) is \( f'(g(x)) \cdot g'(x) \).

In the context of the given exercise, we can consider \( y = \frac{\ln x}{\ln 4} \) as a product of a constant \( \frac{1}{\ln 4} \) and a natural logarithm \( \ln x \). The chain rule allows us to differentiate efficiently by treating the constant separately, resulting in \( \frac{1}{\ln 4} \cdot \frac{1}{x} \). The chain rule simplifies the differentiation of more complicated functions into manageable steps, using the derivative of the inside function and the outside function.