Problem 15 - Calculus and its applications

Question

Differentiate. $$ y=\log _{17} x $$

Step-by-Step Solution

Answer

\( \frac{1}{x \cdot \ln(17)} \) is the derivative of \( y = \log_{17}(x) \).

Step 1: Understand the Logarithm Function

The function given is a logarithm with base 17. The general formula for changing the base of a logarithm to the natural logarithm is: \[ \log_b(x) = \frac{\ln(x)}{\ln(b)} \] In this case, the function can be rewritten as: \[ y = \frac{\ln(x)}{\ln(17)} \].

Step 2: Apply Derivative Rules

To differentiate \( y = \frac{\ln(x)}{\ln(17)} \), recognize that \( \ln(17) \) is a constant. Therefore, the differentiation of \( y \) with respect to \( x \) becomes applying the constant multiplier rule on \( \ln(x) \): \[ \frac{dy}{dx} = \frac{1}{\ln(17)} \cdot \frac{d}{dx}[\ln(x)] \] Since the derivative of \( \ln(x) \) is \( \frac{1}{x} \), we have: \[ \frac{dy}{dx} = \frac{1}{\ln(17)} \cdot \frac{1}{x} \].

Step 3: Simplify the Derivative

Combine the terms to complete the differentiation: \[ \frac{dy}{dx} = \frac{1}{x \cdot \ln(17)} \].

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is an invaluable tool, especially when dealing with functions that are either complex products or quotients, or involve exponentials or powers. In simple terms, this technique involves using logarithms to simplify the differentiation process.
When we have a function that is expressed as the logarithm of a variable, like \( y = \log_{17}(x) \), our first step is to transform this into a form that is easier to differentiate. This is where changing the base of the logarithm to the natural logarithm (base \( e \)) comes in handy using the formula:\[\log_b(x) = \frac{\ln(x)}{\ln(b)}\]
In our specific problem, changing the base of the logarithm converts the function into:\[y = \frac{\ln(x)}{\ln(17)}\]Now, it is much easier to apply the standard rules of differentiation.

Natural Logarithm

The natural logarithm, represented by \( \ln(x) \), is a special kind of logarithm with a base of \( e \), where \( e \) is approximately equal to 2.71828. It is the inverse operation of taking the exponential function to that base.
The function of the natural logarithm is widely used in calculus because of its properties that simplify differentiation and integration.

• The derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \).
• This simple derivative makes logarithms very handy when differentiating more complex functions that involve products, quotients, or powers.

The use of natural logarithms allows us to apply rules and simplify expressions, as seen in the exercise where the base 17 logarithm is converted to a quotient involving \( \ln(x) \). This conversion simplifies the differentiation process, leveraging the constant \( \ln(17) \) to make calculations straightforward.

Derivative Rules

Differentiating functions requires the application of various derivative rules. In calculus, these rules simplify finding the rate of change of functions.
In our exercise, we encounter a couple of derivative rules:

• Constant Multiple Rule: If \( c \) is a constant and \( f(x) \) is a differentiable function, then the derivative of \( c \cdot f(x) \) is \( c \cdot f'(x) \). This rule applies when we differentiate \( y = \frac{\ln(x)}{\ln(17)} \), recognizing \( \frac{1}{\ln(17)} \) is a constant.
• Derivative of Natural Logarithm: Knowing that the derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \), assists in simplifying the solution.

Putting these together, the derivative \( \frac{dy}{dx} \) becomes\[\frac{1}{\ln(17)} \cdot \frac{1}{x} = \frac{1}{x \cdot \ln(17)}\]These rules streamline the differentiation process, presenting a clear pathway from the function to its derivative.