Problem 19 - Calculus with Concepts in Calculus

Question

Calculate the logarithm by using \((11)\). $$ \log _{\pi} e $$

Step-by-Step Solution

Answer

\( \log_{\pi} e \approx 0.873 \).

Step 1: Identify the base and the number

In this problem, we need to calculate the logarithm of \( e \) with base \( \pi \). This is written as \( \log_{\pi} e \).

Step 2: Use the change of base formula

The change of base formula allows us to evaluate logarithms with any base by converting them into fractions of logarithms with common bases (such as base 10 or base \( e \)). The formula is \[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \] where \( c \) is commonly an easier base such as 10 or \( e \).

Step 3: Choose base \( e \) for simplification

Using base \( e \) simplifies the calculation because we are dealing with \( \log_{\pi} e \). Thus, we apply: \[ \log_{\pi} e = \frac{\ln e}{\ln \pi} \].

Step 4: Calculate \( \ln e \) and \( \ln \pi \)

The natural logarithm of \( e \), denoted as \( \ln e \), is 1 because \( e^{1} = e \). Calculate \( \ln \pi \) using a calculator, which is approximately 1.1447.

Step 5: Compute the result

Substitute the values obtained into the expression: \[ \log_{\pi} e = \frac{1}{1.1447} \approx 0.873 \]. Thus, \( \log_{\pi} e \approx 0.873 \).

Key Concepts

Change of Base Formula

The change of base formula is a versatile tool that aids in evaluating logarithms with non-standard bases. If you face a logarithm like \( \log_{a} b \) and prefer to convert it into something more manageable, this formula comes to your rescue. To apply the change of base formula, you'll employ this equation:- \[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \]Here, the base \( c \) is chosen for simplicity. - Common choices are base 10 (the common logarithm) or base \( e \) (the natural logarithm). - This conversion facilitates computations, especially when calculators or software can directly compute common or natural logarithms. By converting \( \log_{\pi} e \) with base \( e \), we leverage natural logarithms to ease out calculations smoothly, demonstrating the power of the change of base formula.

Natural Logarithm

Among the various types of logarithms, the natural logarithm, denoted as \( \ln \), holds a special place. It's the logarithm to the base \( e \), where \( e \) is an irrational number approximately equal to 2.71828.Natural logarithms are widely used in many fields such as:- Mathematics,- Physics,- Engineering and,- Economics.Key characteristics of natural logarithms include:- \( \ln e = 1 \) because \( e^1 = e \).- It simplifies calculations in calculus, as we'll see.When dealing with expressions like \( \log_{\pi} e \), using \( \ln \) allows us to leverage these attributes of natural logarithms:- For \( \ln e \), you simply use the fact that it's 1.- For other values like \( \ln \pi \), you'd use a calculator or software to approximate the value.

Calculus Steps

Understanding logarithmic problems often involves identifying the base, determining how to simplify it, and carefully executing each step. This process reflects a methodical approach that can be seen in calculus.Here's how calculus steps apply to solving \( \log_{\pi} e \):- **Identify Components:** Recognize the base and the number, in this case, \( e \) and \( \pi \).- **Apply Simplification Techniques:** Utilizing the change of base formula was our simplification.- **Calculate Key Values:** Compute relevant natural logarithms like \( \ln e \) and \( \ln \pi \). By walking through calculus-inspired steps:- First, realize \( \ln e = 1 \).- Then calculate \( \ln \pi \) using a calculator, approximately 1.1447.- Use these values in the formula \( \log_{\pi} e = \frac{1}{1.1447} \) leading to the simplified result. This meticulous approach ensures you understand each element of the calculation clearly, enhancing your problem-solving skills in mathematical settings.