Problem 72 - Precalculus

Question

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

Step-by-Step Solution

Answer

True, both sides equal 1.

Step 1: Understanding the Expression

The expression we need to determine the truth value of is \( e = \frac{1}{\log 10}\). The natural logarithm \(\ln\) is based on the number \(e\), meaning that \(\ln e = 1\). We should verify if \(\frac{1}{\log 10}\) also equals 1.

Step 2: Calculating the Right-Hand Side Value

Let's calculate \(\frac{1}{\log 10}\). The \(\log\) function without any specified base is typically the common logarithm, meaning base 10. Therefore, \(\log_{10} 10 = 1\). Substituting back, we get \(\frac{1}{\log 10} = \frac{1}{1} = 1\).

Step 3: Comparison of Values

Now, we compare both sides of the equation. On the left side, \(\ln e = 1\), and on the right side, \(\frac{1}{\log 10} = 1\). As both sides equal 1, the given statement \(\ln e = \frac{1}{\log 10}\) holds true.

Key Concepts

Natural logarithm

The natural logarithm, typically denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is approximately 2.71828, a special number called Euler's number. The natural logarithm of a number \( x \) is the power to which \( e \) must be raised to equal \( x \). For example, \( \ln e = 1 \) because \( e^1 = e \). This property makes natural logarithms particularly useful in calculus and complex calculations involving exponential growth or decay.

• Natural logarithms are unique because they simplify a wide array of mathematical problems.
• They are often used in situations involving natural growth processes.

The natural logarithm is widely used in scientific fields due to its natural properties and ease of integration into differential equations. Understanding \( \ln e = 1 \) is fundamental because it highlights the intuitive baseline where the exponential function equals its rate of growth, a concept prevalent in many natural phenomena.

Logarithm properties

Logarithms have some essential properties that make them straightforward to work with in mathematical computations:

• Product Property: \( \log_b(xy) = \log_bx + \log_by \) - This shows that the logarithm of a product is the sum of the logarithms.
• Quotient Property: \( \log_b\left(\frac{x}{y}\right) = \log_bx - \log_by \) - The logarithm of a quotient is the difference of the logarithms.
• Power Property: \( \log_b(x^k) = k \cdot \log_bx \) - The logarithm of a power is the exponent times the logarithm of the base.

These properties help simplify complex calculations by breaking them down into more manageable parts. In the problem given, the property that \( \log_{10}10 = 1 \) was used. This is derived from the basic idea that any number raised to the power of 1 is itself.

Common logarithm

The common logarithm, often just written as \( \log \), refers specifically to a logarithm of base 10. It's used in various applications, particularly when dealing with decimal numbers. The base 10 logarithm is intuitive in everyday life since our numerical system is base 10.

• The common logarithm of 10 is 1, so \( \log_{10} 10 = 1 \).
• Common logarithm is valuable in scientific fields like chemistry and biology where pH scales and concentrations are concerned.
• It is also found in sound intensity level calculations due to the decibel being a base 10 logarithmic scale.

In the problem solved, understanding \( \log 10 = 1 \) helped validate the statement. This connection illustrates the idea that common logarithms provide a practical way to simplify calculations, especially when data ranges over several orders of magnitude.