Problem 77 - Calculus Single Variable

Question

Use logarithmic differentiation to calculate the derivative of the given function. $$ \log _{2}^{x}(x) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{x \cdot \ln(2)} \).

Step 1: Identify the Function Using Logarithms

The function given is \( \log_2^x(x) \). This can be rewritten in terms of natural logarithms (base \(e\)) using the change of base formula: \( \log_b(a) = \frac{\ln a}{\ln b} \). Hence, \( \log_2^x(x) = \frac{\ln(x)}{\ln(2)} \).

Step 2: Apply Logarithmic Functions for Differentiation

Since we have a composition of logs, consider the function as \( y = \log_2^x(x) = \frac{\ln(x)}{\ln(2)} \). Begin by expressing \( y = \ln(x) \cdot \frac{1}{\ln(2)} \), treating \( \ln(2) \) as a constant factor. Thus, the derivative of \( y \) with respect to \( x \) can be written using the derivative of natural log property.

Step 3: Differentiate Using Basic Derivative Rules

Differentiate \( y = \ln(x) \cdot \frac{1}{\ln(2)} \). Since \( \ln(2) \) is a constant, factor it out: \( \frac{d}{dx} \left( \frac{1}{\ln(2)} \ln(x) \right) = \frac{1}{\ln(2)} \cdot \frac{d}{dx}(\ln(x)) \). The derivative of \( \ln(x) \) is \( \frac{1}{x} \) which results in \( \frac{1}{\ln(2)} \cdot \frac{1}{x} \).

Step 4: Simplify the Derivative Expression

Simplify the result: \( \frac{1}{\ln(2)} \cdot \frac{1}{x} = \frac{1}{x \cdot \ln(2)} \). Thus, the derivative of \( \log_2^x(x) \) is \( \frac{1}{x \cdot \ln(2)} \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique for finding derivatives, particularly useful when dealing with functions that have variable bases or exponents, like in the exercise given. This method utilizes the properties of logarithms to simplify the calculation of derivatives.
The core idea is to take the natural logarithm of both sides of an equation before differentiating. By doing so, you transform products into sums and help simplify powers as well. This can make differentiation more straightforward.
For instance, if you have a function like \( y = f(x)^{g(x)} \), you can apply logarithmic differentiation as follows:

• Start by taking the natural logarithm of both sides: \( \ln(y) = \ln(f(x)^{g(x)}) \).
• Use the logarithm power rule: \( \ln(y) = g(x) \cdot \ln(f(x)) \).
• Differentiate both sides with respect to \( x \). You'll apply the chain rule on the left side and the product rule on the right.

This process can considerably simplify solving complex differentiation problems, especially when dealing with logarithmic, exponential, or trigonometric components all in one function.

Natural Logarithm

The natural logarithm is a logarithm with the base \(e\), where \(e\) is approximately 2.71828. It is denoted as \(\ln\).
Natural logs are widely used in calculus due to their unique mathematical properties and how they relate to rates of growth and decay.

• When you differentiate \( \ln(x) \), you get \( \frac{1}{x} \). This differentiation result makes it particularly handy when using logarithmic differentiation.
• Natural logs convert products into sums: \( \ln(ab) = \ln(a) + \ln(b) \).
• They also change powers into products: \( \ln(a^b) = b\ln(a) \). This property is central to simplifying expressions involving exponentials.

In the solution, we used natural logarithms to rewrite \( \log_2^x(x) \) in a simpler form, through using the change of base formula. The derivative then becomes easier to compute as it reduces the necessity of dealing with complicated expressions directly, simplifying \( \ln(x) \cdot \frac{1}{\ln(2)} \).

Change of Base Formula

The change of base formula is a fundamental concept in logarithms. It allows you to convert logarithms from one base to another, which can simplify calculations.
The formula is expressed as:

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

This expression tells us that a logarithm with base \(b\) can be rewritten as a ratio of natural logarithms.
In practice, this is particularly useful when calculators or certain algorithms primarily support only base-\(e\) logarithms. Finding derivatives also becomes more manageable when expressing logarithms in natural log form.

• For instance, \( \log_2^x(x) = \frac{\ln(x)}{\ln(2)} \), simplifying the derivative calculation to recognize \( \ln(2) \) as a constant factor.
• This conversion enables us to utilize the simpler differentiation rules of natural logs, making the problem-solving process more efficient and reducing potential errors.

Understanding and applying this formula is a valuable tool in calculus, greatly aiding in tasks of simplification and calculation of derivatives across various logarithmic and exponential expressions.