Problem 59 - 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. $$\log _{7} 2.61$$

Step-by-Step Solution

Answer

\( \log_7 2.61 \approx 0.492978 \)

Step 1: Identify the Change of Base Formula

The Change of Base Formula allows us to evaluate logarithms with bases other than 10 or e by converting them to ones with those bases. The formula is given by: \[ \log_b a = \frac{\log_k a}{\log_k b} \]where \( b \) is the original base, \( a \) is the argument of the logarithm, and \( k \) is the new base, either 10 (common log) or \( e \) (natural log).

Step 2: Choose the Base for Conversion

For this calculation, we will use the common logarithm (base 10). This choice simplifies use with most calculators and is a common choice for base conversion.

Step 3: Apply the Change of Base Formula

Using the Change of Base Formula, convert the logarithm:\[ \log_7 2.61 = \frac{\log 2.61}{\log 7} \]

Step 4: Calculate the Logarithms

Using a calculator, find the values of the logarithms:- \( \log 2.61 \approx 0.416640 \)- \( \log 7 \approx 0.845098 \)

Step 5: Divide to Find the Result

Divide the calculated values to find the logarithm in base 7:\[ \frac{\log 2.61}{\log 7} = \frac{0.416640}{0.845098} \approx 0.492978 \]

Step 6: Round the Result

Round the answer to six decimal places: \( \log_7 2.61 \approx 0.492978 \).

Key Concepts

Understanding Logarithms

Logarithms are a mathematical concept that helps us find the power needed to raise a specific base to get a given number. If you think of exponentiation as multiplication's counterpart, then logarithms are the inverse operation, like division is to multiplication. We use the notation \( \log_b a \), which represents the exponent needed for the base \( b \) to produce the number \( a \).

• The expression \( \log_b a = x \) is equivalent to saying that \( b^x = a \).
• Logarithms with different bases might arise, such as \( \log_2 \), \( \log_3 \), etc.

Logarithms allow us to solve problems involving exponential growth or decay, making them especially useful in fields like science, engineering, and finance.
When dealing with logarithms involving bases other than 10 or \( e \), we often use the Change of Base Formula to convert them into more manageable forms, using either common or natural logarithms to make calculations easier with standard calculators.

Common Logarithms

Common logarithms are a very special type of logarithms that use 10 as the base. They are often used in real-world applications like sound measurements and human perception of brightness or loudness, which tend to scale logarithmically. You'll frequently encounter common logarithms in mathematics, typically denoted by \( \log \).
In most calculators and mathematical software, the \( \log \) button is designed to calculate common logarithms. This is immensely helpful because measuring the direct logarithm of uncommon bases is tricky.To use common logarithms for the Change of Base Formula in the given exercise, you can express the base 7 logarithm as:

• \( \log_7 2.61 = \frac{\log 2.61}{\log 7} \)

By using base 10 in a calculator, you can efficiently evaluate logarithms that are not naturally suited for manual computation. It's a practical method to accomplish otherwise challenging calculations with ease.

Natural Logarithms

Natural logarithms use the mathematical constant \( e \) (approximately 2.718) as their base. We denote natural logarithms with \( \ln \), and they arise naturally when dealing with continuous growth or decay processes, such as radioactive decay, population growth, or calculating compound interest.
While common logarithms (base 10) are frequently used in practical applications, natural logarithms excel in calculus and advanced mathematics because they relate simplistically to exponential functions.In the context of the Change of Base Formula, the natural logarithm can also be used to convert a logarithm to a different base:

• For example, \( \log_7 2.61 = \frac{\ln 2.61}{\ln 7} \)

Using the natural logarithm button on your calculator facilitates such transformations, especially when tackling equations involving exponential and logarithmic relationships. Understanding both common and natural logarithms equips you with versatile tools to solve a variety of mathematical challenges.