Problem 36 - Technical Mathematics with Calculus

Question

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$-3.846$$

Step-by-Step Solution

Answer

\( \log(x) \approx \frac{-3.846}{\ln(10)} \approx -1.6707 \)

Step 1: Understand the Concept of Change of Base

The change of base formula allows us to convert a logarithm from one base to another. For logarithms, the common logarithm has base 10, denoted as \( \log(x) \), and the natural logarithm has base \( e \), denoted as \( \ln(x) \). The formula to change from \( \ln(x) \) to \( \log(x) \) is given by \( \log(x) = \frac{\ln(x)}{\ln(10)} \) because \( \log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)} \) for any positive bases \( b \) and \( c \) that are not equal to 1.

Step 2: Apply the Change of Base Formula

Given the natural logarithm of a number is \( -3.846 \), we use the change of base formula to find its common logarithm: \( \log(x) = \frac{\ln(x)}{\ln(10)} = \frac{-3.846}{\ln(10)} \)

Key Concepts

Common Logarithm

The common logarithm is an essential concept in mathematics, particularly when dealing with powers of 10. A common logarithm, denoted as \( \log(x) \), is essentially a logarithm with a base of 10. This means that if you have \( 10^n = x \), then the common logarithm \( \log(x) \) is simply the exponent \( n \).

For example, if we consider the number 100, which is \( 10^2 \), the common logarithm \( \log(100) \) is 2. Understanding this is crucial when working with large numbers in scientific notation, as it can significantly simplify calculations.

• Base 10 is assumed if no base is written explicitly.
• The common logarithm of 10 is always 1, because \( 10^1 = 10 \).
• The common logarithm of 1 is always 0, because \( 10^0 = 1 \).

This indicates the power to which 10 must be raised to yield a certain number. The common logarithm hence becomes very useful in various fields including, engineering, sciences, and finance.

Natural Logarithm

In contrast to the common logarithm, the natural logarithm uses the mathematical constant \( e \) (approximately equal to 2.71828) as its base and is represented as \( \ln(x) \). The natural logarithm is central to growth processes, such as compound interest, population dynamics, and the spread of diseases.

When you see \( e^n = x \), the natural logarithm \( \ln(x) \) is the exponent \( n \). This logarithm is aptly named 'natural' because it appears routinely across natural phenomena and in mathematical functions like differentiation and integration.

• The natural logarithm of \( e \) is always 1, because \( e^1 = e \).
• For any positive number \( x \), \( e^{\ln(x)} = x \).

The relation between natural logarithms and common logarithms is fundamental, as it allows conversions between different bases through the change of base formula.

Logarithmic Conversion

The ability to convert between different bases of logarithms is important, especially when working with calculators or mathematical models that utilize specific logarithmic bases. This is where the change of base formula comes into play. It enables mathematicians to convert a logarithm of one base to another, for example, converting natural logarithms to common logarithms and vice versa.

To apply this in practice, as demonstrated in the exercise provided, if the natural logarithm of a number is \( -3.846 \), and we wish to find its common logarithm (base 10), we employ the change of base formula:
\[ \log(x) = \frac{\ln(x)}{\ln(10)} \]
By substituting the given natural logarithm into this formula, you could determine the equivalent common logarithm. This conversion process is not only critical for computations but also for understanding the relationships between different types of logarithmic functions.

Application in Calculations

Consider the scenario where a graphing calculator only has a button for natural logarithms (\( \ln \)). To find the common logarithm of a number, you would use this formula to convert it. This illustrates the practical significance of the logarithmic conversion, making it easier to solve logarithmic problems when tools are limited to certain logarithmic functions.