Problem 65 - Algebra and Trigonometry

Question

Use the Change of Base Formula to show that $$\log e=\frac{1}{\ln 10}$$

Step-by-Step Solution

Answer

\(\log_{10} e = \frac{1}{\ln 10}\).

Step 1: Recall the Change of Base Formula

The change of base formula states that for any positive numbers \(a\), \(b\), and \(c\), where \(a eq 1\), \(b eq 1\), and \(c > 0\), the logarithm \(\log_b c\) can be rewritten as \(\frac{\log_a c}{\log_a b}\). This formula allows us to change the base of a logarithm to another base.

Step 2: Identify Specific Values

For this problem, we want to rewrite \(\log_{10} e\) using the change of base formula. Here, \(c = e\), \(b = 10\), and we'll use the natural logarithm with base \(e\), so \(a = e\).

Step 3: Apply the Change of Base Formula

Using the change of base formula: \(\log_{10} e = \frac{\ln e}{\ln 10}\). Since the natural logarithm (ln) of a number is the logarithm with base \(e\), \(\ln e = 1\).

Step 4: Simplify the Expression

Substitute \(\ln e = 1\) into the expression: \(\log_{10} e = \frac{1}{\ln 10}\). Thus, the original expression simplifies directly to \(\frac{1}{\ln 10}\).

Key Concepts

Logarithms

Logarithms are an essential mathematical concept that help us understand and calculate how many times we need to multiply one number, called the base, to get another number. For example, in the logarithm \(log_b c\), "\(b\)" is the base, and "\(c\)" is the number we aim to achieve through repeated multiplication.
This can be expressed as \(b^x = c\), where \(x\) is the logarithm of \(c\) with base \(b\), written as \(log_b c = x\). This relationship is particularly useful in computer science, statistics, and various fields of engineering.

Some key things to note about logarithms include:

• The logarithm of 1 with any base is always 0, because any number raised to the power of 0 equals 1.
• A logarithm becomes undefined if the base is negative or 1, as these situations result in contradictions.

To simplify calculations across different bases, we often use the change of base formula, which provides a method for converting logarithms to a different base, as shown in the attached exercise.

Natural Logarithm

The natural logarithm, denoted as \(ln\), is a special type of logarithm where the base is the constant \(e\), approximately equal to 2.71828. This constant is essential in many areas of mathematics, especially in calculus and exponential growth equations. Its unique properties make it especially useful for continuous compounding, population growth, and in solving differential equations.

Natural logarithms have certain unique traits:

• The natural logarithm of \(e\) itself, \(ln e\), is 1, because \(e^1 = e\).
• Natural logarithms are particularly useful when dealing with exponential growth processes.
• They simplify the derivatives and integrals of exponential functions, making them crucial in advanced mathematics.

In the exercise above, the natural logarithm plays a key role when utilizing the change of base formula, to show that \(\log_{10} e = \frac{1}{\ln 10}\), by substituting \(\ln e = 1\).

Base Conversion

Base conversion is a method used to change the base of a logarithm to a base that is more convenient for calculation or further analysis. This is made possible with the change of base formula, which takes a logarithm \(log_b c\) and allows us to express it in terms of a new base \(a\), using the formula:
\[log_b c = \frac{\log_a c}{\log_a b}\].
This adaptability simplifies calculations when certain bases suit a problem better—like converting to base \(e\) for calculus-related problems using natural logarithms.

Some important notes about base conversion include:

• It allows switching between common logarithmic bases, such as 10 and \(e\).
• It can simplify complex logarithmic expressions, as demonstrated in the original exercise.

The change of base formula allows for flexibility, as seen when expressing \(log_{10} e\) in terms of natural logarithms. This conversion results in \(\frac{1}{\ln 10}\), demonstrating the power of base conversion in mathematical calculations.