Problem 71 - Precalculus

Question

Determine whether each statement is true or false. $$\log e=\frac{1}{\ln 10}$$

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understanding the problem

We need to determine if the statement \( \log e = \frac{1}{\ln 10} \) is true or false. Here, \( \log \) refers to the base 10 logarithm (common logarithm), and \( \ln \) is the natural logarithm with base \( e \).

Step 2: Expressing \( \log e \) in terms of \( \ln e \)

The property of logarithms allows us to express common log in terms of natural log: \( \log a = \frac{\ln a}{\ln 10} \). So, \( \log e = \frac{\ln e}{\ln 10} \).

Step 3: Simplifying \( \log e \)

Since the natural logarithm of \( e \) (i.e., \( \ln e \)) is 1, our expression becomes: \( \log e = \frac{1}{\ln 10} \).

Step 4: Comparing Expressions

The given statement \( \log e = \frac{1}{\ln 10} \) matches with the derived expression \( \log e = \frac{1}{\ln 10} \) from the earlier step. Thus, the statement is true.

Key Concepts

Common Logarithm

The common logarithm is a very handy tool in mathematics. When you hear "common logarithm," think base 10. It's written as \( \log a \). Why base 10? Because it's practical and easy to use in real-world applications. For example, our decimal system uses base 10. This makes common logarithms intuitive in scientific calculations.
When you calculate the common logarithm of a number, you’re finding out what power 10 must be raised to in order to get that number. For example, \( \log 100 = 2 \), because 10 raised to the power of 2 is 100. It's straightforward, right?
Let's remember that turning a natural log into a common log involves a little trick with the number \( e \), the base of natural logarithms. The relationship is easily expressed as:

• \( \log a = \frac{\ln a}{\ln 10} \)

This is absolutely crucial to know when you're converting between different bases.

Natural Logarithm

The natural logarithm is another type of logarithm that differs from the common one. Represented as \( \ln a \), it uses the mathematical constant \( e \) (approximately 2.71828) as its base. The term "natural" comes from its frequent appearance in natural growth processes, like population growth and radioactive decay.
Understanding \( \ln \) becomes easier when you think of it this way: it's about finding the power you need to raise \( e \) to get the number \( a \). For instance, \( \ln e = 1 \), because \( e^1 = e \).
Natural logarithms pop up in many areas of calculus because \( e \) has unique properties. When you're solving equations or analyzing functions, they frequently make life simpler. This is why it's important to understand their properties and behavior.

Logarithmic Properties

Logarithmic properties allow you to simplify complex expressions and solve equations more readily. These properties are fundamental in understanding and manipulating logarithms.
Here are some essential properties:

• Product Property: \( \log_b (mn) = \log_b m + \log_b n \)
• Quotient Property: \( \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \)
• Power Property: \( \log_b (m^n) = n \cdot \log_b m \)

These properties hold true across different bases, whether it's the common logarithm or the natural logarithm.
In the specific exercise we tackled, knowing that \( \log e = \frac{\ln e}{\ln 10} = \frac{1}{\ln 10} \) used the property of converting logarithms between bases. These foundational principles make handling logarithmic expressions much simpler, allowing quick verification of equations like \( \log e = \frac{1}{\ln 10} \).