Problem 80 - Thomas Calculus in SI Units

Question

Find the derivative of \(y\) with respect to the given independent variable. $$y=3 \log _{8}\left(\log _{2} t\right)$$

Step-by-Step Solution

Answer

\( \frac{dy}{dt} = \frac{3}{t \cdot \log_{e} 8 \cdot \log_{e} 2 \cdot \log_{e} t} \).

Step 1: Identify the Structure

We have the function \( y = 3 \log_{8} (\log_{2} t) \). This is a composite logarithmic function, meaning we will need to apply the chain rule and change of base formula to differentiate it.

Step 2: Apply Change of Base Formula

Recall the change of base formula for logarithms: \( \log_{b} x = \frac{\log_{a} x}{\log_{a} b} \). Apply this to both parts of the function: \( \log_{8} (\log_{2} t) = \frac{\log_{e} (\log_{2} t)}{\log_{e} 8} \) and \( \log_{2} t = \frac{\log_{e} t}{\log_{e} 2} \).

Step 3: Simplify the Inner Function

Substitute \( \log_{2} t = \frac{\log_{e} t}{\log_{e} 2} \) into this expression and rewrite: \( \log_{8} (\log_{2} t) = \frac{\log_{e} (\frac{\log_{e} t}{\log_{e} 2})}{\log_{e} 8} \).

Step 4: Differentiate Using Chain Rule

Rewrite \( g(t) = \log_{e} (\frac{\log_{e} t}{\log_{e} 2}) \). The derivative \( g'(t) = \frac{1}{(\frac{\log_{e} t}{\log_{e} 2})} \cdot \frac{1}{t} \cdot \frac{1}{\log_{e} 2} \), using chain rule and derivative of log.

Step 5: Put It All Together

Now, differentiate \( y = 3 \cdot \frac{g(t)}{\log_{e} 8} \). Applying chain rule: \( \frac{dy}{dt} = 3 \cdot \frac{1}{\log_{e} 8} \cdot \frac{1}{\frac{\log_{e} t}{\log_{e} 2}} \cdot \frac{1}{t} \cdot \frac{1}{\log_{e} 2} \). Simplify: \( \frac{dy}{dt} = \frac{3}{t \cdot \log_{e} 8 \cdot \log_{e} 2 \cdot \log_{e} t} \).

Key Concepts

Understanding Composite Functions

A composite function is essentially a function within another function.
In other words, you have one function applied to the result of another function. The expression you might see often looks like this:

• \( f(g(x)) \)

Composite functions require special handling when differentiating since you need to account for both layers. In our exercise, the function is \( y = 3 \log_{8}(\log_{2} t) \).
Here, the inner function is \( \log_{2} t \), and the outer function is \( \log_{8} \). To find the derivative, you apply the chain rule, which involves differentiating the outer function first and multiplying it by the derivative of the inner function. This systematic approach ensures you correctly handle the layers of the composite function.

Applying the Change of Base Formula

The change of base formula is a useful tool for simplifying logarithms with bases other than 10 or \( e \).
It's expressed as:

• \( \log_{b} x = \frac{\log_{a} x}{\log_{a} b} \)

This means you can rewrite a logarithm with any base \( b \) as a fraction of natural logarithms or logarithms of another convenient base. This conversion is crucial in our exercise to separate the logarithms into more manageable parts using the natural log base \( e \).
For instance, we used this formula in both \( \log_{8}(\log_{2} t) \) and \( \log_{2} t \) to facilitate easier differentiation. Once rewritten, the calculus becomes more straightforward, as you deal with the derivatives of natural logarithms.

Derivative of Logarithmic Functions

Logarithm derivatives are quite distinct from ordinary functions. The trick here is to remember the base and how it affects the differentiation process. For a natural logarithm \( \log_{e}(x) \), the derivative is simply \( \frac{1}{x} \).
If you have a different base, like our exercise with \( \log_{2}(x) \), you have to incorporate the natural log of that base, resulting in:

• The derivative formula: \( \frac{1}{x \cdot \log_{e} b} \)

In our function, differentiating \( \log_{e}(\frac{\log_{e} t}{\log_{e} 2}) \) required a thoughtful approach via the chain rule.
By recognizing and applying the correct derivative formulas for logarithms, the problem resolves into simpler sequences that can be managed step by step. Understanding these rules lets you confidently tackle the derivatives of more complex logarithmic functions.