Question

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$-0.638$$

Short answer

The common logarithm of the number is approximately -0.277.

Step-by-step solution

Understand the Relationship Between Natural and Common Logarithms

The natural logarithm, often denoted as ln(x), is a logarithm to the base e. The common logarithm, also known as the base-10 logarithm or simply log(x), uses base 10. To convert from a natural logarithm to a common logarithm, you can use the change of base formula which is \(\log_b{a} = \frac{\ln{a}}{\ln{b}}\), where a is the value you have the natural logarithm of and b is the new base you want to convert to, in this case, base 10.

Apply the Change of Base Formula

To find the common logarithm of the number with the given natural logarithm (-0.638), apply the change of base formula: \(\log{a} = \frac{\ln{a}}{\ln{10}}\). Since you're given \(\ln{a} = -0.638\), plug this into the formula to get \(\log{a} = \frac{-0.638}{\ln{10}}\).

Calculate the Common Logarithm

Use a calculator to find \(\ln{10}\), which is approximately 2.3026. Then divide the given natural logarithm (-0.638) by \(\ln{10}\) to find the common logarithm. The calculation will be \(\log{a} = \frac{-0.638}{2.3026}\) which equals approximately -0.277.

Key concepts

Natural Logarithm

The natural logarithm is a mathematical function often denoted by \( \ln(x) \). It's called 'natural' because of its inherent appearance in nature and mathematics, especially in problems involving growth and decay patterns like those found in biology, chemistry, and physics. The base of a natural logarithm is the irrational number \(e\), approximately equal to 2.71828. This number has unique properties that make it ideal as the foundation of continuous growth calculations.

Understanding the natural logarithm is crucial, especially when dealing with exponential functions. It answers the question: 'To what power must we raise \(e\) in order to get \(x\)?'. So when you see \(\ln(x)\), it represents the power that \(e\) is raised to obtain the number \(x\).

Practical applications of the natural logarithm span various fields including compound interest calculations in finance, calculating population growth rates in biology, and even in algorithms used in computer science.

Common Logarithm

When we mention the common logarithm, we refer to logarithms with a base of 10, represented as \( \log(x) \). In contrast to \(\ln(x)\), which uses an irrational number as its base, \(\log(x)\) uses the familiar base of 10 and is especially useful in the context of scaling and scientific notation because our number system is decimal (base-10).

The common logarithm asks the question: 'To what power must 10 be raised to yield the number \(x\)?' This function is frequently used in science and engineering fields to manage the vast range of values, like when measuring the intensity of earthquakes with the Richter scale or understanding sound levels in decibels.

Historically, before calculators were widely available, common logarithms were used to simplify calculations in navigation, engineering, and other practical fields by turning complex multiplication into manageable addition.

Logarithmic Conversion

Logarithmic conversion takes a value from one logarithmic base to another. It's a powerful tool when you need to switch the 'language' of logarithms to make the mathematics easier or more understandable in certain contexts. The change of base formula is the key to making this conversion. It is expressed as:
\[\log_b{a} = \frac{\ln{a}}{\ln{b}}\]
where \(a\) is the value whose logarithm you know in one base, and you wish to express it in terms of \(b\), the new base. This formula essentially leverages the relationship between different logarithmic bases to evaluate logs in terms of bases that are more convenient or relevant to the problem at hand.

In practice, since calculators typically only have buttons for natural and common logarithms, the change of base formula is particularly useful. It empowers us to calculate logs to any base using just these two functions. This concept underscores the universality and interconnectedness of logarithmic functions across different bases and their applications in real-world problem-solving situations.