Problem 34 - College Algebra

Question

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. $$ \log _{8}(65) $$

Step-by-Step Solution

Answer

\( \log_{8}(65) \approx 2.00779 \)

Step 1: Understanding the Change of Base Formula

The change-of-base formula allows us to convert a logarithm of any base to a different base, often to base 10 or base e (natural logarithm), which are standard on calculators. The formula is: \[ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \]For this exercise, we'll use base 10 for our calculations. So we need to find \( \log_{10}(65) \) and \( \log_{10}(8) \).

Step 2: Applying the Formula

Using the change-of-base formula, we apply it specifically to our example:\[ \log_{8}(65) = \frac{\log_{10}(65)}{\log_{10}(8)} \]This converts our logarithm with base 8 into a quotient of logarithms with base 10.

Step 3: Calculating Logarithms

Now, use a calculator to find each of the logarithms to five decimal places.1. Calculate \( \log_{10}(65) \). The approximate value is 1.81291.2. Calculate \( \log_{10}(8) \). The approximate value is 0.90309.

Step 4: Computing the Quotient

Using the calculated values:\[ \log_{8}(65) = \frac{1.81291}{0.90309} \]Execute this division using a calculator to get the final result.

Step 5: Final Result

After dividing the two results, we find:\[ \log_{8}(65) \approx 2.00779 \]This is the value of our log expression evaluated using the change-of-base formula.

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics, used to solve equations where the unknown appears as the exponent of some number. They are particularly useful in dealing with exponential growth or decay processes. A logarithm answers the question, "To what power must we raise a certain base to obtain a given number?"

For example, in the expression \( \log_b(a) \), "b" is the base, "a" is the result, and the logarithm itself represents the exponent that base "b" must be raised to acquire "a". Logarithms can significantly simplify calculations and can transform multiplicative processes into additive ones, which is a key feature in many mathematical and practical applications.

Logarithms have different bases, but the most common are base 10, known as common logarithms, and base \( e \), known as natural logarithms. Calculators usually have dedicated buttons for these common bases, which makes calculations straightforward.

Natural Log

The natural log, denoted as \( \ln(x) \), is a logarithm with the base \( e \), where \( e \approx 2.71828 \). Natural logarithms are fundamental in calculus and many areas of science because \( e \) itself is a critical constant in mathematics due to its unique properties related to growth rates.

Using \( \ln \) helps simplify expressions involving exponential growth, decay, and many calculus operations. When solving problems involving logarithms, converting them into natural logs can sometimes simplify the process, especially when dealing with continuous growth scenarios.

It is crucial to be comfortable with natural logs as they often appear in advanced mathematical contexts and various applied sciences. Conveniently, because they are based on \( e \), they align well with naturally occurring processes, making them highly relevant for real-world applications.

Base Conversion

Base conversion in logarithms is a powerful technique that allows us to switch from one logarithmic base to another. This is particularly useful when a calculator does not readily support the initial base of the logarithm we are dealing with.

To convert a logarithm with an arbitrary base "b" to base "c" using the change of base formula, we apply:

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

This formula helps in simplifying and solving logarithmic expressions by converting difficult-to-calculate bases into more manageable ones, such as base 10 or natural base \( e \), which most calculators can handle.

In practice, using the change of base formula allows us to compute logarithmic expressions in various mathematics problems, finance applications, and computer science, where different scales and bases might be initially inconvenient.

Calculator Usage

Using a calculator for logarithms is essential, especially in complex calculations involving non-standard bases. Most scientific calculators come with buttons for \( \log \) (base 10) and \( \ln \) (natural log), enabling users to efficiently handle logarithmic calculations.

To calculate a logarithm, you typically need to:

• Identify the base of the logarithm
• Use the change of base formula if necessary, converting to base 10 or \( e \)
• Enter the logarithmic value into the calculator with the correct button
You can quickly find the quotient of natural logs by dividing the values calculated through the change of base formula on your calculator.

Besides providing approximations to several decimal places, calculators also help ensure precision, especially with intricate math problems where exact values are needed, such as calculating compounded interest, solving rate equations, or simply checking manual logarithmic computations.