Problem 58 - Algebra A Combined Function

Question

Approximate each logarithm to four decimal places. $$ \log _{6} \frac{2}{3} $$

Step-by-Step Solution

Answer

Approximately -0.2263.

Step 1: Understand the Logarithm

You are asked to find the logarithm of a fraction to the base 6, specifically \( \ log_{6} \frac{2}{3} \). Logarithms can transform fractions into subtractive expressions.

Step 2: Apply the Logarithm Quotient Rule

According to the logarithm quotient rule: \[ \log_{b} \left( \frac{M}{N} \right) = \log_{b}(M) - \log_{b}(N) \].In this problem, this becomes:\[ \log_{6} \left( \frac{2}{3} \right) = \log_{6}(2) - \log_{6}(3) \].

Step 3: Convert to Base 10 Logarithms

Since base-6 logarithms are not easily calculated without a calculator, we convert them into base 10 using the formula \( \log_{b}(x) = \frac{\log_{10}(x)}{\log_{10}(b)} \).This means we need to compute:\[\log_{6}(2) = \frac{\log_{10}(2)}{\log_{10}(6)} \and \ \log_{6}(3) = \frac{\log_{10}(3)}{\log_{10}(6)} \].

Step 4: Calculate the Base 10 Logarithms

Using a calculator:- \(\log_{10}(2) \approx 0.3010\)- \(\log_{10}(3) \approx 0.4771\)- \(\log_{10}(6) \approx 0.7782\).

Step 5: Perform the Division

Now divide the results from Step 4:\[ \log_{6}(2) = \frac{0.3010}{0.7782} \approx 0.3868 \\log_{6}(3) = \frac{0.4771}{0.7782} \approx 0.6131 \].

Step 6: Subtract the Logarithms

Finally, calculate the difference from Step 5 as per Step 2:\[ \log_{6}(\frac{2}{3}) = 0.3868 - 0.6131 \= -0.2263 \].

Step 7: Write the Final Answer

The value of \( \log_{6}(\frac{2}{3}) \) is approximately -0.2263, rounded to four decimal places.

Key Concepts

Base Conversion

When working with logarithms, being able to convert between different bases is a crucial skill. This is particularly important because certain base logarithms (like base-6) are not directly available on standard calculators. Base conversion helps us to express these logarithms in terms of a base that is more familiar and easy to deal with, such as base-10.
To convert a logarithm from one base to another, we often use the conversion formula:

• \( \log_{b}(x) = \frac{\log_{10}(x)}{\log_{10}(b)} \)

This means you're finding the common logarithm (base-10) of both the subject \( x \) and the original base \( b \), and then taking the ratio of the two.
This method allows us to perform calculations even if the calculator does not support direct computation for non-standard bases, making it a versatile tool in your mathematical toolkit.

Logarithm Quotient Rule

The logarithm quotient rule is a property of logarithms that simplifies the computation of the logarithm of a fraction. This rule states:

• \( \log_{b} \left( \frac{M}{N} \right) = \log_{b}(M) - \log_{b}(N) \)

What this means is that when you have a logarithm of a fraction, you can break it down into two separate logs of the numerator and denominator, and then subtract the second from the first.
This property is particularly useful because it allows a complex expression to be dealt with in smaller, more manageable parts, streamlining the problem-solving process for logarithmic equations.

Approximation

When dealing with logarithms, it is often necessary to approximate values to make calculations manageable, especially when working without exact base functions available. For instance, in the given exercise, the final result is approximated to four decimal places.
Approximating logarithms typically involves:

• Using a calculator or log tables to find values to a certain level of precision,
• Rounding the calculated values to the required number of decimal places after applying operations like subtraction or division.

The objective of approximation is to provide a number that is close enough to the exact answer for practical purposes, without significantly altering the utility or the outcome of the expression being calculated.

Mathematical Calculation

Mathematical calculation of logarithms involves several steps that require precision and attention to detail. In exercises like the one given, each step builds upon the previous:

• First, apply mathematical rules (such as the quotient rule) to simplify the expression,
• Use base conversion to express logarithms in a solvable form, like base-10,
• Perform necessary divisions of the logarithm values,
• Finally, subtract the adjusted logarithmic values as instructed by the problem.

A detailed and methodical approach ensures accuracy when working with logarithms, allowing complex problems to be solved step by step, ultimately yielding precise approximations.