Problem 33 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to the given independent variable. \(y=\log _{5} e^{x}\)

Step-by-Step Solution

Answer

The derivative is \(\frac{1}{\ln(5)}\).

Step 1: Express in Terms of Natural Logarithms

The function given is a logarithm with base 5. We know that a logarithm with any base can be expressed in terms of natural logarithms (ln) using the formula: \[y = \log_{b}(a) = \frac{\ln(a)}{\ln(b)}\]Thus, for the function \(y = \log_{5} e^{x}\), we can write:\[ y = \frac{\ln(e^{x})}{\ln(5)}\]

Step 2: Simplify the Natural Log Expression

Using the property of natural logarithms that \(\ln(e^{x}) = x\cdot \ln(e)\), and knowing that \(\ln(e) = 1\), simplify the expression:\[y = \frac{x \cdot \ln(e)}{\ln(5)} = \frac{x}{\ln(5)}\]

Step 3: Differentiate with Respect to x

The derivative of \(y = \frac{x}{\ln(5)}\) with respect to \(x\) involves differentiating a constant multiplied by \(x\). Since \(\ln(5)\) is a constant, the derivative is:\[\frac{dy}{dx} = \frac{1}{\ln(5)} \cdot \frac{d}{dx}(x) = \frac{1}{\ln(5)}\]

Step 4: State the Final Derivative

The derivative of the function \(y = \log_{5} e^{x}\) with respect to \(x\) is therefore:\[\frac{dy}{dx} = \frac{1}{\ln(5)}\]

Key Concepts

Logarithms

Understanding logarithms is essential in mathematics, especially when dealing with exponential terms. At its core, a logarithm is the inverse of exponentiation. For instance, if you have the equation \( b^x = y \), the logarithm with base \( b \) tells you what power \( x \) the base \( b \) was raised to get \( y \). Written mathematically, this is \( x = \log_b(y) \). Logarithms are particularly useful for simplifying multiplication into addition. This is because \( \log_b(A \times B) = \log_b(A) + \log_b(B) \). Additionally, logs make working with powers more straightforward with the rule \( \log_b(A^n) = n \times \log_b(A) \). This feature of logarithms provides an invaluable way to simplify and solve exponential equations. To work with logarithms effectively, it is vital to remember these properties and recognize scenarios where rewriting an equation using logs will make solving it much simpler.

Natural Logarithm

The natural logarithm is a specific type of logarithm that uses the base \( e \), where \( e \approx 2.71828 \). It is denoted as \( \ln(x) \). This constant \( e \) is an irrational number, and it is fundamental in calculus due to its natural properties in continuous growth processes. To comprehend why \( \ln(x) \) is so useful, consider its relationship with the exponential function. The natural logarithm \( \ln(x) \) serves as the inverse of the exponential function \( e^x \). As such, this relationship is crucial: if you know \( e^y = x \), then \( \ln(x) = y \). Natural logarithms simplify many aspects of calculus, especially with derivatives and integrals. For example, the derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \). This property is extremely beneficial, especially when transforming complex multiplicative relationships in functions into simpler additive forms, which are much easier to differentiate or integrate.

Change of Base Formula

The change of base formula is an essential tool for converting logarithms from one base to another. This is particularly useful when you are limited to calculation methods that only support natural logs or common logs (base 10). The change of base formula states:

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

This means that any logarithm can be expressed in terms of natural logarithms. For example, according to the solution to our exercise, turning \( \log_5(e^x) \) into \( \frac{\ln(e^x)}{\ln(5)} \) involved applying the change of base formula. This formula is advantageous because it allows you to use logarithm properties with ease across various different bases, making calculations more flexible. Additionally, by converting to natural logarithms, we can leverage their beneficial calculus properties, such as easier differentiation and integration. To master the change of base formula, practice applying it to different types of logarithmic expressions. Over time, this will enhance your ability to transition seamlessly between logarithmic bases in different mathematical contexts.