Problem 60 - Algebra A Combined Function

Question

Approximate each logarithm to four decimal places. $$ \log _{1 / 3} 2 $$

Step-by-Step Solution

Answer

The approximation is -0.6309.

Step 1: Convert to Natural Logarithms

Using the change of base formula, we convert the logarithm to a more computable form:\[ \log_{1/3} 2 = \frac{\ln 2}{\ln (1/3)} \]

Step 2: Calculate Natural Logarithm of 2

Approximate the natural logarithm of 2:\[ \ln 2 \approx 0.6931 \]

Step 3: Calculate Natural Logarithm of 1/3

Approximate the natural logarithm of 1/3:\[ \ln \left(\frac{1}{3}\right) \approx -1.0986 \]

Step 4: Divide Logarithms

Divide the natural logarithm of 2 by the natural logarithm of 1/3:\[ \frac{0.6931}{-1.0986} \approx -0.6309 \]

Step 5: Finalize the Approximation

The approximate value of \( \log_{1/3} 2 \) to four decimal places is \(-0.6309\).

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool in mathematics. It allows us to simplify the calculation of logarithms with bases other than 10 or the natural number e. By converting a logarithm to a different base, we manage to compute it using calculators or lookup tables more easily. The formula is given by:

• \( \log_b a = \frac{\log_k a}{\log_k b} \)

Here, \( b \) represents the base we start with, and \( k \) can be any base we prefer, like 10 or e (natural logarithm). This transformation helps make complicated base calculations manageable. In our exercise, we switched \( \log_{1/3} 2 \) to use the natural logarithm base with \( \ln \).

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with a special base—Euler's number, \( e \), approximately 2.71828. In mathematical computations and calculus, \( \ln \) is often preferred because of its natural occurrence in various scientific contexts such as exponential growth and continuous compounding.
This makes the natural logarithm vital in many scientific and engineering fields. For the exercise, we converted \( \log_{1/3} 2 \) using the natural logarithm, giving us:\

• \( \ln 2 \approx 0.6931 \)
• \( \ln (1/3) \approx -1.0986 \)

The negative sign in \( \ln (1/3) \) indicates that \( 1/3 \) is less than 1, a key aspect to understand when working with logarithms.

Approximation

Approximation is the process of finding a value that is close enough to the correct value, within an acceptable error margin. It's crucial in logarithmic calculations, especially since most logarithms do not have a simple, exact form, like the natural logarithm of 2 or \( 1/3 \).
In our example, we approximate these up to four decimal places:

• \( \ln 2 \approx 0.6931 \)
• \( \ln (1/3) \approx -1.0986 \)

When dividing these approximations, we obtain the final approximation \( -0.6309 \), which keeps our results both practical and accurate. Approximations are valuable for making complex math concepts more comprehensible and relevant to real-world applications.

Base Conversion

Base conversion involves changing the base in which a logarithmic operation is performed. It's a technique that ensures calculations are simpler and more universally applicable. In our example, converting the base from \( \log_{1/3} \) to \( \ln \) allowed us to use a more computable form of logarithms, given by:

• \( \frac{\ln 2}{\ln (1/3)} \)

This change in base makes use of values and tools readily available on calculators, which are limited to certain bases like 10 and \( e \).
Adopting a different base is particularly useful when we deal with non-standard values or work with algebraic expressions in broader scientific computations, making difficult problems easier to manage.