Problem 109 - Precalculus

Question

Refer to the following: In calculus, to find the derivative of a function of the form \(y=k^{x}\) where \(k\) is a constant, we apply logarithmic differentiation. The first step in this process consists of writing \(y=k^{x}\) in an equivalent form using the natural logarithm. Use the properties of this section to write an equivalent form of the following implicitly defined functions. $$y=2^{x}$$

Step-by-Step Solution

Answer

The equivalent form is \( \ln(y) = x \cdot \ln(2) \).

Step 1: Express as Natural Logarithm

Start by taking the natural logarithm of both sides of the function \(y = 2^x\). When you take the logarithm, it becomes \( \ln(y) = \ln(2^x) \).

Step 2: Apply Logarithm Power Rule

Use the power rule of logarithms, which states that \( \ln(a^b) = b \cdot \ln(a) \), to simplify \( \ln(2^x) = x \cdot \ln(2) \). Therefore, the equation becomes \( \ln(y) = x \cdot \ln(2) \).

Step 3: Conclude with Equivalent Form

Thus, the function \(y = 2^x\) can be rewritten in an equivalent form as \( \ln(y) = x \cdot \ln(2) \). This form is useful for differentiating the function implicitly.

Key Concepts

Natural Logarithm

A natural logarithm is a logarithm with the base equal to Euler's number, denoted as \( e \), which is approximately 2.718. This type of logarithm is commonly used in calculus because it simplifies many expressions and makes differentiation easier. When we take the natural logarithm of a number or an expression, we are simply using the notation \( \ln \). For instance, when taking the natural logarithm of a power expression like \( 2^x \), we write it as \( \ln(2^x) \).

• The natural logarithm helps convert multiplication into addition, which is handy in calculus.
• It is particularly useful in exponential growth models and decay processes, where calculations could otherwise become cumbersome.

In the context of logarithmic differentiation, the natural log allows us to handle multiplicative functions with ease, transforming them into a form that is more manageable for derivation.

Power Rule of Logarithms

The power rule of logarithms is a crucial property that aids in simplifying logarithmic expressions, especially when working with exponents. According to this rule, the logarithm of a power \( a^b \) can be expressed as \( b \cdot \ln(a) \). Hence, \( \ln(a^b) = b \cdot \ln(a) \). This rule is fundamental when we need to differentiate functions involving powers.

• The power rule allows us to bring the exponent in front as a multiplier, which simplifies the expression considerably for differentiation.
• It reduces the complexity of expressions before differentiating, making the process straightforward.

In the example given, by applying the power rule to \( \ln(2^x) \), we simplify it to \( x \cdot \ln(2) \). This step is critical in transforming functions into a form ready for differentiation.

Implicit Differentiation

Implicit differentiation is a technique used when it is difficult to solve for \( y \) explicitly before differentiating. Instead of solving for \( y \), we differentiate the entire equation with respect to \( x \), treating \( y \) as a function of \( x \). This is especially useful in cases like exponential functions where direct differentiation isn't straightforward.

• Implicit differentiation allows us to manage more complex relationships between variables.
• Because \( y \) is treated as a dependent variable, its derivatives are represented as \( \frac{dy}{dx} \).

After using logarithmic differentiation to express a function like \( y = 2^x \) as \( \ln(y) = x \cdot \ln(2) \), we differentiate implicitly. This involves differentiating both sides of the equation with respect to \( x \), allowing us to find \( \frac{dy}{dx} \) without isolating \( y \) first.