Problem 81 - Thomas Calculus in SI Units

Question

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

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{t} \).

Step 1: Rewrite the Logarithm Expression

Start by rewriting the given function with a simpler base for the logarithm. Remember that you can express the logarithm in base 2 as a division of logarithms in base e (natural log):\[ y = \frac{\ln(8t^{\ln 2})}{\ln 2} \]

Step 2: Apply Logarithmic Properties

Use the properties of logarithms to simplify the expression inside the natural log. The logarithmic property \( \ln(a^b) = b\ln(a) \) can be applied:\[ \ln(8t^{\ln 2}) = \ln 8 + \ln(t^{\ln 2}) = \ln 8 + (\ln 2) \ln t \]

Step 3: Simplify the Expression

Substitute the simplified expression from the previous step back into the original expression for \( y \):\[ y = \frac{\ln 8}{\ln 2} + \frac{(\ln 2) \ln t}{\ln 2} \]Now, simplify the expression:\[ y = 3 + \ln t \]

Step 4: Differentiate the Function

Now that \( y \) is simplified, differentiate it with respect to \( t \). The derivative of a constant is zero and the derivative of \( \ln t \) is \( \frac{1}{t} \):\[ \frac{dy}{dt} = 0 + \frac{1}{t} = \frac{1}{t} \]

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique to differentiate complex functions involving products, quotients, and powers, particularly when variables are both in the base and the exponent. In the context of our exercise, the original function involves a complex term inside a logarithm: \( \log_{2}(8t^{\ln 2}) \). This makes logarithmic differentiation an ideal approach.

• Why Use Logarithmic Differentiation? Instead of directly differentiating a complicated function, we can take the logarithm on both sides to simplify the expression.
• This often converts multiplication into addition, division into subtraction, and powers into coefficients, making differentiation easier.
• After simplifying using logarithms, we differentiate the resulting simpler expression using standard derivative rules.

By converting the base of the logarithm to a natural logarithm (base \( e \)), the differentiation becomes more manageable. You then simplify the expression inside the log using logarithmic properties, which will be discussed next.

Properties of Logarithms

The properties of logarithms provide tools to simplify expressions, especially when dealing with products, quotients, and exponents. These properties are essential when working through complex functions like the one in the exercise: \( \ln(8t^{\ln 2}) \).

Key Properties Used:

• Product Rule: \( \ln(a \cdot b) = \ln a + \ln b \) - This allows us to separate the product into a sum.
• Power Rule: \( \ln(a^b) = b \cdot \ln a \) - This lets us move an exponent in front of the logarithm, turning a power into a coefficient.

For our example, we used these properties to simplify \( \ln(8t^{\ln 2}) \) into \( \ln 8 + (\ln 2) \ln t \). By applying these rules, the differentiation becomes straightforward.These properties convert exponential and multiplicative structures into additive and simpler linear forms, facilitating easier differentiation. This is crucial because the simplicity after applying the properties determines the ease with which you can find the derivative.

Natural Logarithms

Natural logarithms are logarithms with base \( e \) (where \( e \approx 2.71828 \)), and are a standard in calculus due to their unique properties that simplify calculations, especially when dealing with differentiation.

• Why Use Natural Logarithms: They are universally applicable in calculus for functions that involve growth and decay like exponential functions.
• The derivative of \( \ln x \) is particularly simple, \( \frac{d}{dx} \ln x = \frac{1}{x} \), which is advantageous in differentiation problems.

In the given step-by-step solution, changing our base 2 logarithm to a natural logarithm helped utilize these simplifications. After simplification, the function was reduced to \( y = 3 + \ln t \), making it straightforward to find the derivative.Ultimately, the natural logarithms reduce complexity and enhance the functionality of logarithmic differentiation, paving the path to an easier differentiation process.