Problem 25 - College Algebra

Question

Rewrite each expression as an equivalent ratio of logs using the indicated base. $$ \log _{7}(15) \text { to base } e $$

Step-by-Step Solution

Answer

\( \log_{7}(15) = \frac{\ln(15)}{\ln(7)} \) with base \( e \).

Step 1: Identify the Change of Base Formula

To rewrite a logarithm in terms of a different base, we can use the change of base formula: \[ \log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)} \]Here, \( a = 15 \), \( b = 7 \), and \( c = e \) (the natural logarithm, indicated by \( \ln \)).

Step 2: Apply the Change of Base Formula

Using the change of base formula, rewrite the logarithmic expression:\[ \log_{7}(15) = \frac{\ln(15)}{\ln(7)} \]This ratio of natural logs is the equivalent expression with base \( e \).

Key Concepts

Logarithm

A logarithm is essentially the opposite of raising a number to a power. In other words, it answers the question: "To what exponent must we raise a certain base to obtain a given number?" For example, in the expression \(log_{b}(a)\), \(b\) is the base, and \(a\) is the number we're interested in. The logarithm tells us the power to which \(b\) must be raised to equal \(a\).
Additionally, logarithms help simplify the process of dealing with large numbers, especially in the context of multiplying, dividing, and handling powers. This is due to their useful properties:

• Adding logs corresponds to multiplying their arguments: \(log_{b}(m) + log_{b}(n) = log_{b}(m \times n)\)
• Subtracting logs aligns with division: \(log_{b}(m) - log_{b}(n) = log_{b}\left(\frac{m}{n}\right)\)
• The logarithm of a power: \(log_{b}(m^n) = n \times log_{b}(m)\)

These properties make logarithms a valuable tool in various fields, including mathematics, engineering, and the sciences.

Equivalent Ratio of Logs

An equivalent ratio of logs allows one to express a logarithm in terms of a different base. This is possible through the Change of Base Formula. This formula states that for any positive numbers \( a \) and \( b \) where \( b eq 1 \), and any new base \( c \) where \( c eq 1 \):
\[\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}\]
This equation shows that by using a common logarithmic base \( c \), you can calculate logarithms of any base. The Change of Base Formula is especially handy when calculators or computational tools have specific base log functions readily available, like base 10 and the natural logarithm \( \ln \).
For example, if you wanted to find \( \log_{7}(15) \) using base \( e \) (which is the natural log), you would use:
\[\log_{7}(15) = \frac{\ln(15)}{\ln(7)}\]
This means you are converting the base from 7 to \( e \), making it easier to compute with the tools at your disposal.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a specific logarithm where the base is the constant \( e \) (approximately 2.71828). The number \( e \) is significant in mathematics, particularly in calculus and complex analysis, because it serves as the base for exponential growth and decay processes.
To say \( \ln(a) \) means "the power to which \( e \) must be raised to equal \( a \)." It is widely used across various fields due to its unique mathematical properties and the natural occurrence of \( e \) in continuous growth patterns.
The natural logarithm has several key properties:

• \( \ln(1) = 0 \) because \( e^0 = 1 \)
• \( \ln(e) = 1 \) since \( e^1 = e \)
• \( \ln(e^x) = x \) for any real number \( x \)

These properties make the natural logarithm indispensable in problems involving continuous change, such as calculating compound interest, analyzing population dynamics, and solving differential equations.