Question

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$3.775$$

Short answer

\(\log(a) \approx \frac{3.775}{\ln(10)}\approx 1.6387\)

Step-by-step solution

Understand the Relationship Between Common Logarithm and Natural Logarithm

The common logarithm is a logarithm with base 10 and is represented by \(\log(x)\). The natural logarithm has base \(e\) and is represented by \(\ln(x)\). The relationship between them for a number \(a\) is given by the change of base formula: \(\log(a) = \frac{\ln(a)}{\ln(10)}\).

Apply the Change of Base Formula

To find the common logarithm of the number whose natural logarithm is 3.775, apply the change of base formula: \[\log(a) = \frac{\ln(a)}{\ln(10)}\] Here, \(\ln(a) = 3.775\). Now substitute the given natural logarithm value into the formula: \[\log(a) = \frac{3.775}{\ln(10)}\].

Calculate the Common Logarithm

Use a calculator to find the value of \(\ln(10)\), then divide 3.775 by this value to find \(\log(a)\).

Key concepts

Understanding the Common Logarithm

The common logarithm is a key concept in mathematics, particularly in the field of logarithms. It is defined as the logarithm with base 10, commonly denoted as \( \log(x) \). When you see a logarithm written without a base, such as \( \log(100) \), it's assumed to be a common logarithm, meaning \( \log_{10}(100) \).

The common logarithm has unique properties that make it particularly useful for calculations and real-world applications. For instance, \( \log(10^n) = n \), which simplifies the process of finding the order of magnitude of a number, important in fields such as science and engineering.

In classroom settings and standardized tests, understanding the common logarithm is essential as it frequently appears in exercises involving exponential growth, pH in chemistry, and decibel levels in physics. It is a fundamental tool in representing large or small numbers in a manageable form.

The Power of the Natural Logarithm

Contrasting the common logarithm, which uses base 10, the natural logarithm uses the irrational number \(e\) as its base, which is approximately 2.71828. Denoted by \( \ln(x) \), the natural logarithm holds a special place in mathematics due to its natural occurrence in various contexts, such as compound interest, population growth, and even in describing certain physical phenomena.

The value of \(e\) arises naturally in the process of calculating continuous growth and decay rates, making the natural logarithm an indispensable tool for modeling real world scenarios in economics and biology. \( \ln(x) \), like \( \log(x) \), is also a key player in calculus, particularly when dealing with integrations and derivatives of exponential functions.

Exploring Logarithmic Relationships

Logarithmic relationships enable us to understand how different logarithms relate to each other, which is crucial for various mathematical operations and applications. A profound aspect of logarithms is their ability to transform products into sums and quotients into differences, making complex multiplications and divisions simpler to manage.

One such critical relationship is the change of base formula: \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \), which shows how to convert between any two logarithmic bases. This formula pops up frequently not just in mathematical exercises, but also in algorithm complexities in computer science and signal processing's decibels calculations.

Underlying these relationships is the power of logarithms to simplify expressions and solve equations that would otherwise be very difficult or impossible to crack with traditional algebra. By mastering logarithmic relationships, students can unravel a wide array of problems with more elegance and less effort.