Problem 52 - Precalculus

Question

Evaluate the logarithms using the change-of-base formula. Round to four decimal places. $$\log _{5} \frac{1}{2}$$

Step-by-Step Solution

Answer

\(\log_5 \frac{1}{2} \approx -0.4306\).

Step 1: Understanding the Change-of-Base Formula

The change-of-base formula allows us to evaluate logarithms with any base by converting them into common logarithms (base 10) or natural logarithms (base e). The formula is: \(\log_b a = \frac{\log_c a}{\log_c b}\) where \(c\) is a new base, commonly 10 or \(e\).

Step 2: Applying the Change-of-Base Formula

We want to find \(\log_5 \frac{1}{2}\). Using the change-of-base formula, we can use base 10: \(\log_5 \frac{1}{2} = \frac{\log_{10} \frac{1}{2}}{\log_{10} 5}\).

Step 3: Calculating Logarithms with Base 10

Now calculate \(\log_{10} \frac{1}{2}\) and \(\log_{10} 5\) using a calculator. \(\log_{10} \frac{1}{2} \approx -0.3010\) and \(\log_{10} 5 \approx 0.6990\).

Step 4: Computing the Result

Divide the results from the previous step to find the value of \(\log_5 \frac{1}{2}\): \(\log_5 \frac{1}{2} = \frac{-0.3010}{0.6990} \approx -0.4306\).

Key Concepts

Logarithms

Logarithms are fundamental in mathematics, especially in dealing with exponential equations. At its core, a logarithm answers the question: "To what power must a specific base be raised, to achieve a certain value?" For instance, the logarithm \(\log_b a\) represents the exponent needed for the base \(b\) to equal \(a\). This is often expressed in the equation \(b^x = a\), where \(x = \log_b a\). The ability to transform exponentiation into straightforward multiplication and division is what makes logarithms incredibly handy.

• Inverse function: Logarithms act as the inverse of exponentials. While an exponential function grows rapidly, logarithms allow us to manage large numbers.
• Base flexibility: Logarithms can have different bases, such as 2, 5, 10, or even the natural base \(e\) (approximately 2.71828).

This fundamental concept supports a variety of applications in fields such as engineering, science, and finance. Understanding how to manipulate and evaluate them is crucial for various mathematical problems.

Base 10

Base 10 logarithms, also known as "common logarithms," are widely used due to their ease of use and the prevalence of the decimal system in everyday mathematics. In notation, base 10 logarithms are typically denoted by \(\log_{10} a\), but are often just written as \(\log a\) when the base doesn’t need emphasis due to its commonness.

• Historical significance: Historically, base 10 was used extensively for simplifying complex calculations before calculators became common.
• Practical applications: They are particularly useful in applications related to scientific notation and data analysis.

Base 10 logarithms are advantageous because they simplify multiplication into addition. For example, they convert products into sums, making it easier to handle numbers on a logarithmic scale. This is why they are an integral part of our mathematical toolkit, especially when utilizing the change-of-base formula, which allows computation of logarithms with bases other than 10 by converting them to base 10.

Natural Logarithms

Natural logarithms are special types of logarithms with base \(e\), where \(e\) is a mathematical constant approximately equal to 2.71828. The notation for natural logarithms is typically \(\ln\), instead of \(\log_e\), to signify they are with base \(e\). These are particularly important because of their relationships to growth processes.

• Connection to calculus: Natural logarithms frequently arise in calculus, especially in calculations involving growth rates, such as in compounded interest and population modeling.
• Properties of \(e\): The constant \(e\) is unique due to properties of exponential growth, making natural logarithms crucial for understanding continuous growth models.

Natural logarithms simplify many mathematical expressions involving exponentials with base \(e\). They are used in various scientific and engineering disciplines to model growth problems, decay processes, and in solving differential equations, making them invaluable for scientific computation.