Compute Log Base 6 Of 1/9 Using The Change Of Base Formula

Introduction

In mathematics, logarithms are a fundamental concept, representing the inverse operation to exponentiation. They are used extensively in various fields, including science, engineering, and finance. However, calculators often only compute logarithms in base 10 (common logarithm) or base e (natural logarithm). When we need to calculate a logarithm with a different base, such as base 6 in this case, we employ the change of base formula. This article provides a comprehensive guide on how to use the change of base formula to compute $\log_6 \frac{1}{9}$, and rounds the answer to the nearest thousandth. Understanding this method will empower you to solve a wider range of logarithmic problems.

Understanding Logarithms

Before diving into the calculation, let's briefly review what logarithms are. The logarithm of a number x to the base b is the exponent to which b must be raised to produce x. Mathematically, this is expressed as:

$\log_b x = y$ if and only if $b^y = x$

Where:

• b is the base of the logarithm (b > 0, b ≠ 1)
• x is the argument (the number whose logarithm is being computed, x > 0)
• y is the exponent or the logarithm itself

For instance, $\log_{10} 100 = 2$ because $10^2 = 100$. Similarly, $\log_2 8 = 3$ because $2^3 = 8$. Understanding this foundational concept is crucial for applying the change of base formula effectively.

Common Logarithms and Natural Logarithms

Two logarithmic bases are particularly important:

1. Common Logarithm: This is the logarithm with base 10, denoted as $\log_{10}$ or simply $\log$. Calculators typically have a $\log$ button for computing common logarithms.
2. Natural Logarithm: This is the logarithm with base e (Euler's number, approximately 2.71828), denoted as $\log_e$ or $\ln$. Calculators usually have an $\ln$ button for computing natural logarithms.

Since most calculators are equipped to compute base 10 and base e logarithms, the change of base formula becomes essential for calculating logarithms with other bases.

The Change of Base Formula

The change of base formula allows us to convert a logarithm from one base to another. This formula is particularly useful when we need to calculate a logarithm with a base that our calculator does not directly support. The formula is stated as follows:

$\log_b a = \frac{\log_c a}{\log_c b}$

Where:

• a is the argument of the logarithm
• b is the original base
• c is the new base (typically 10 or e for calculator use)

This formula indicates that the logarithm of a to the base b is equal to the logarithm of a to the base c divided by the logarithm of b to the base c. By choosing c as either 10 or e, we can use standard calculators to compute logarithms with any base.

Applying the Change of Base Formula: A Step-by-Step Guide

To effectively use the change of base formula, follow these steps:

1. Identify the Original Base and Argument: Determine the original base (b) and the argument (a) of the logarithm you want to compute. In our case, we want to compute $\log_6 \frac{1}{9}$, so b = 6 and a = $\frac{1}{9}$.
2. Choose a New Base: Select a new base (c) that your calculator can handle. The most common choices are base 10 (common logarithm) or base e (natural logarithm). We can use either, and the result will be the same. For this example, we'll use both to illustrate.
3. Apply the Formula: Substitute the values into the change of base formula:
   • Using base 10: $\log_6 \frac{1}{9} = \frac{\log_{10} \frac{1}{9}}{\log_{10} 6}$
   • Using base e: $\log_6 \frac{1}{9} = \frac{\ln \frac{1}{9}}{\ln 6}$
4. Compute the Logarithms: Use a calculator to find the values of the logarithms in the numerator and the denominator.
5. Divide: Divide the logarithm of the argument by the logarithm of the original base.
6. Round: Round the result to the desired decimal places. In this case, we need to round to the nearest thousandth.

Calculating $\log_6 \frac{1}{9}$ Using the Change of Base Formula

Let's apply the change of base formula to compute $\log_6 \frac{1}{9}$.

Step 1: Identify the Original Base and Argument

The original base, b, is 6. The argument, a, is $\frac{1}{9}$.

Step 2: Choose a New Base

We'll use both base 10 (common logarithm) and base e (natural logarithm) to demonstrate the equivalence.

Step 3: Apply the Formula

• Using base 10: $\log_6 \frac{1}{9} = \frac{\log_{10} \frac{1}{9}}{\log_{10} 6}$
• Using base e: $\log_6 \frac{1}{9} = \frac{\ln \frac{1}{9}}{\ln 6}$

Step 4: Compute the Logarithms

First, we recognize that $\frac{1}{9}$ is the same as $9^{-1}$, so $\log_{10} \frac{1}{9} = \log_{10} 9^{-1} = -\log_{10} 9$. Using a calculator, we find:

• $\log_{10} 9 \approx 0.9542$
• $\log_{10} \frac{1}{9} \approx -0.9542$
• $\log_{10} 6 \approx 0.7782$

Using natural logarithms:

• $\ln \frac{1}{9} \approx -2.1972$
• $\ln 6 \approx 1.7918$

Step 5: Divide

• Using base 10: $\frac{\log_{10} \frac{1}{9}}{\log_{10} 6} \approx \frac{-0.9542}{0.7782} \approx -1.2262$
• Using base e: $\frac{\ln \frac{1}{9}}{\ln 6} \approx \frac{-2.1972}{1.7918} \approx -1.2262$

Step 6: Round

Rounding to the nearest thousandth, we get -1.226.

Therefore, $\log_6 \frac{1}{9} \approx -1.226$.

Common Mistakes and How to Avoid Them

When using the change of base formula, it's crucial to avoid common errors that can lead to incorrect results. Here are some typical mistakes and how to steer clear of them:

1. Incorrectly Applying the Formula: One common mistake is confusing the numerator and denominator in the change of base formula. Ensure you correctly place the logarithm of the argument in the numerator and the logarithm of the original base in the denominator. Double-check the formula before computing.
2. Calculator Errors: Using the calculator incorrectly, such as misentering numbers or not using the correct logarithm function (e.g., using the common logarithm instead of the natural logarithm), can lead to errors. Always verify the input and the function used.
3. Rounding Errors: Rounding intermediate values too early can introduce significant errors in the final result. It's best to keep as many decimal places as possible during the calculation and only round the final answer to the required precision.
4. Forgetting the Negative Sign: When dealing with fractions or negative exponents, it’s easy to overlook the negative sign. Pay close attention to the signs of the logarithms and ensure they are correctly carried through the calculation.
5. Misunderstanding the Logarithm Concept: A lack of clear understanding of what logarithms represent can lead to confusion and errors. Ensure you have a solid grasp of the logarithmic concept and its properties before applying the change of base formula.

By being mindful of these common pitfalls, you can improve your accuracy and confidence when working with logarithms and the change of base formula.

Practical Applications of Logarithms and the Change of Base Formula

Logarithms and the change of base formula aren't just theoretical concepts; they have numerous practical applications in various fields:

1. Science:
   • Chemistry: Logarithms are used to express pH values, which measure the acidity or alkalinity of a solution. The pH scale is logarithmic, making it easier to represent a wide range of concentrations.
   • Physics: Logarithms are used in calculations involving sound intensity (decibels), earthquake magnitude (Richter scale), and radioactive decay.
2. Engineering:
   • Electrical Engineering: Logarithmic scales are used in Bode plots to analyze the frequency response of circuits.
   • Acoustical Engineering: Logarithms are used to measure sound levels and analyze acoustic data.
3. Finance:
   • Compound Interest: Logarithms are used to calculate the time it takes for an investment to grow to a certain value at a given interest rate.
   • Financial Modeling: Logarithmic scales are often used to analyze financial data and model growth rates.
4. Computer Science:
   • Algorithm Analysis: Logarithms are used to describe the efficiency of algorithms, particularly in sorting and searching algorithms.
   • Data Compression: Logarithmic functions are used in data compression algorithms to reduce the amount of storage space required.
5. Mathematics:
   • Solving Exponential Equations: Logarithms are essential for solving exponential equations, where the variable appears in the exponent.
   • Calculus: Logarithmic functions are used extensively in calculus, particularly in integration and differentiation.

Real-World Examples

• Earthquake Magnitude: The Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale. An earthquake of magnitude 7 is ten times stronger than an earthquake of magnitude 6.
• Sound Intensity: The decibel scale, used to measure sound intensity, is also logarithmic. A sound of 60 decibels is ten times louder than a sound of 50 decibels.
• pH Measurement: The pH scale ranges from 0 to 14, with each unit representing a tenfold change in acidity or alkalinity. A solution with a pH of 3 is ten times more acidic than a solution with a pH of 4.

The change of base formula allows scientists, engineers, and other professionals to work with logarithms in different bases, making complex calculations more manageable and providing valuable insights into various phenomena.

Conclusion

The change of base formula is a powerful tool for calculating logarithms with any base, especially when calculators are limited to base 10 and base e. In this article, we demonstrated how to use the formula to compute $\log_6 \frac{1}{9}$, arriving at an approximate answer of -1.226. By understanding the steps involved and being aware of common mistakes, you can confidently apply this formula to solve a variety of logarithmic problems. Moreover, recognizing the practical applications of logarithms in science, engineering, finance, and computer science highlights the importance of mastering this mathematical concept. Keep practicing, and you'll find logarithms and the change of base formula to be valuable assets in your mathematical toolkit.