Question

How can the value of a natural logarithm be converted to the value of a common logarithm? (A) The natural logarithm is divided by a constant. (B) A constant is added to or subtracted from the natural logarithm. (C) The natural logarithm is raised to an exponent. (D) The inverse of the natural logarithm is taken.

Short answer

Option (A) is correct: The natural logarithm is divided by a constant.

Step-by-step solution

Understand the relationship between natural logarithm and common logarithm

The natural logarithm \( \text{ln}(x) \) and the common logarithm \( \text{log}_{10}(x) \) are related by the formula: \[ \text{log}_{10}(x) = \frac{\text{ln}(x)}{\text{ln}(10)} \]. This relationship involves dividing the natural logarithm by a constant.

Identify the correct option

From the given options, the natural logarithm is divided by a constant to convert it to a common logarithm. This matches option (A).

Key concepts

Natural Logarithm

The natural logarithm, often written as \( \text{ln}(x) \), is a logarithmic function with the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. This type of logarithm is commonly used in mathematics, especially in calculus and complex calculations involving growth and decay processes.

The natural logarithm is essential in various fields due to its unique properties. One of them is the ability to simplify multiplication and division of numbers into addition and subtraction of exponents.

For instance, the equation \( e^{\text{ln}(x)} = x \) showcases how the natural logarithm and the exponential function (base \( e \)) are inverses of each other. This can be especially useful in solving equations where the variable is an exponent.

Common Logarithm

The common logarithm, denoted as \( \text{log}_{10}(x) \), is a logarithmic function with a base of 10. This type of logarithm is widely used in many practical applications, including in scientific notation, sound intensity (decibels), and the pH of solutions.

Understanding common logarithms is straightforward due to the base 10, which aligns with the decimal number system we use daily. For example, \( \text{log}_{10}(100) = 2 \) because \( 10^2 = 100 \).

Common logarithms are particularly useful in calculations involving large or very small numbers as they can simplify the process of multiplication and division through their logarithmic properties.

Logarithmic Relationships

Logarithmic relationships reveal how different types of logarithms can be converted and manipulated. There is a key formula that connects the natural logarithm \( \text{ln}(x) \) with the common logarithm \( \text{log}_{10}(x) \):

\[ \text{log}_{10}(x) = \frac{\text{ln}(x)}{\text{ln}(10)} \]

This equation shows that you can convert a natural logarithm to a common logarithm by dividing by \( \text{ln}(10) \). Since \( \text{ln}(10) \) is just a constant (approximately 2.30259), this relationship is very handy.

For example, if you have \( \text{ln}(x) = 1.609 \), you can find \( \text{log}_{10}(x) \) as follows:

\[ \text{log}_{10}(x) = \frac{1.609}{2.30259} \approx 0.698 \]

Understanding these relationships helps in converting between logarithmic bases and solving various logarithmic equations more easily.