Problem 60 - Precalculus Mathematics for Calculus

Question

Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$\log _{6} 532$$

Step-by-Step Solution

Answer

\(\log_6 532 \approx 3.504004\)

Step 1: Identify the Change of Base Formula

The Change of Base Formula for logarithms allows us to express a logarithm in terms of logs of any base we choose. The formula is given by: \[ \log_b a = \frac{\log_c a}{\log_c b} \]We can choose base 10 (common logarithm, denoted \(\log\)) or base \(e\) (natural logarithm, denoted \(\ln\)). For this problem, we'll use common logarithms.

Step 2: Apply the Change of Base Formula

Using the Change of Base Formula with common logarithms, we express \(\log_6 532\) as:\[ \log_6 532 = \frac{\log_{10} 532}{\log_{10} 6} \]This translates the base 6 logarithm into one calculable using a standard calculator.

Step 3: Use a Calculator to Find Values

Calculate \(\log_{10} 532\) using a calculator:\[ \log_{10} 532 \approx 2.726833 \]Next, calculate \(\log_{10} 6\) using a calculator:\[ \log_{10} 6 \approx 0.778151 \]

Step 4: Evaluate Final Expression

Now, divide the two values obtained to find \(\log_6 532\):\[ \log_6 532 = \frac{2.726833}{0.778151} \approx 3.504004 \]This is the value of \(\log_6 532\) rounded to six decimal places.

Key Concepts

Natural Logarithms

Natural logarithms are logarithms with the base of the mathematical constant, e, which is approximately equal to 2.71828. They are represented as \( \ln \). Natural logarithms are widely used in mathematics, particularly in calculus and complex analysis.
Some interesting aspects of natural logarithms include:

• They can simplify calculations involving exponential growth and decay.
• Natural logarithms are often found in formulas describing natural processes.
• The notation \( \ln(x) \) stands for \( \log_e(x) \).

To calculate a natural logarithm using a calculator, you can usually find a button labeled 'ln'. Press it after entering the number for which you need the logarithm.

Common Logarithms

Common logarithms use 10 as their base and are often denoted simply as \( \log \), without specifying the base. These are incredibly useful in various fields such as science and engineering because they align well with our decimal number system.
Some notable points about common logarithms include:

• They simplify working with very large or very small numbers.
• Common logarithms are the default logarithm type in many calculators.
• In scientific notation, powers of 10 are often expressed using common logarithms.

To calculate a common logarithm, look for the 'log' button on your calculator. This allows for straightforward computation of logarithms of base 10.

Logarithm Calculation

Logarithm calculation involves determining the power to which a particular base must be raised to obtain a given number. Using the change of base formula makes this process easier, especially when the base is not common (such as 10 or e).
For example, to compute \( \log_6 532 \), a calculator cannot directly determine the value. Instead, you apply the change of base formula:

• Choose a more common base like 10 or e.
• Compute the logarithm of the number and the base with this common base.
• Divide the two results to find the original logarithm.

This method bridges the gap when direct calculation is challenging, and it is especially helpful for non-standard bases. The result can be entered precisely and rounded as necessary, usually to six decimal places for high precision.

Base Conversion

Base conversion in logarithms is about translating a logarithm from one base to another, which is made simple using the change of base formula. This formula expresses a logarithm in terms of another logarithm with a different base. The general formula is:

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

For convenient calculation, we often convert to base 10 (common logarithms) or base e (natural logarithms). Here’s why base conversion is valuable:

• It allows for better compatibility with calculators and computation systems.
• Helps in solving logarithms that are otherwise not easily computable.
• It provides a way to work consistently with familiar systems and streamline calculations.

Understanding this concept enables smoother transitions between different logarithmic bases, making complex mathematical operations more approachable.