Problem 273 - Precalculus

Question

For the following exercises, 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

\( \log_7(15) = \frac{\ln(15)}{\ln(7)} \)

Step 1: Identify the Requirement

We need to change the base of the logarithm from base 7 to base e (natural logarithm). To do this, we will use the change of base formula.

Step 2: Recall the Change of Base Formula

The change of base formula is \( \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \), where \(c\) is the new base. In this problem, \(a=7\), \(b=15\), and \(c=e\).

Step 3: Apply the Change of Base Formula

Using the change of base formula, we write \( \log_7(15) \) as \( \frac{\ln(15)}{\ln(7)} \). This is because we are changing the base to \(e\), where \( \ln \) represents \( \log_e \).

Step 4: Conclusion

Thus, \( \log_7(15) \) expressed in terms of base \(e\) is \( \frac{\ln(15)}{\ln(7)} \).

Key Concepts

Understanding Logarithms

Logarithms are mathematical operations that help us comprehend relationships in terms of exponents. They are essentially the inverse of exponential functions. For instance, when you have a function like \( a^x = b \), the logarithm helps us find \( x \) given the values for \( a \) and \( b \). Logarithms use different bases:

• Common logarithms have a base of 10 and are written as \( \log_{10} \).
• Natural logarithms have a base of \( e \) (approximately 2.718) and are represented as \( \ln \).
• Other numbers as a base, such as 2 or 7, can also be used.

The purpose of converting between these bases is usually to simplify calculations or align with requirements for specific mathematical contexts. Knowing how to translate between bases using the change of base formula is vital for working with logarithms.

The Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a logarithm with base \( e \). The number \( e \) is a mathematical constant approximately equal to 2.718 and arises frequently in mathematical modeling of growth processes.Natural logarithms have practical implications, especially in fields like biology and finance where growth rates are continuously compounded. Understanding \( \ln \) makes it easier to solve problems involving exponential growth.The notation \( \ln(a) \) means the power to which we must raise \( e \) to get \( a \). For example, if \( \ln(x) = 2 \), this implies \( e^2 = x \). The role of \( \ln \) is crucial in the change of base calculations, allowing conversion from one logarithmic base to another, adding flexibility in evaluating expressions.

Equivalent Expressions and Change of Base Formula

Understanding equivalent expressions is about expressing the same mathematical value in different ways. For logarithms, equivalent expressions allow the interchange of bases.The change of base formula is a tool for converting logarithms to a different base. It is given by: \[ \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \]where \( a \) is the original base, \( b \) is the number in the log argument, and \( c \) is the new base. Applying this formula helps you express any logarithm in terms of another base, often simplifying calculations.For the problem \( \log_{7}(15) \) to be expressed in natural logarithms, we set:

• \( a = 7 \)
• \( b = 15 \)
• \( c = e \)

Using the change of base formula, \( \log_7(15) \) becomes \( \frac{\ln(15)}{\ln(7)} \), a form that can be computed with a scientific calculator or software. This equivalence is crucial for versatility in problem-solving.