Problem 65 - Precalculus with Calculus Previews

Question

In Problems \(57-68\), use \(\log _{b} 4=0.6021\) and \(\log _{b} 5=\) \(0.6990\) to evaluate the given logarithm. Round your answer to four decimal places. $$ \log _{b} 0.8 $$

Step-by-Step Solution

Answer

\( \log_b 0.8 = -0.0969 \).

Step 1: Express 0.8 as a Fraction

To evaluate the logarithm, express 0.8 as a fraction. Notice that 0.8 can be written as \( \frac{4}{5} \).

Step 2: Use Logarithm Quotient Rule

Apply the logarithm quotient rule: \( \log_b \left( \frac{4}{5} \right) = \log_b 4 - \log_b 5 \). This allows us to use the given logarithmic values.

Step 3: Substitute Known Logarithm Values

Substitute the known values into the expression from Step 2: \( \log_b 4 = 0.6021 \) and \( \log_b 5 = 0.6990 \).

Step 4: Calculate and Round the Result

Perform the subtraction: \( 0.6021 - 0.6990 = -0.0969 \). Round the result to four decimal places, which remains unchanged at \( -0.0969 \).

Key Concepts

Logarithm Quotient Rule

The logarithm quotient rule is a useful property of logarithms. It allows you to simplify the logarithm of a fraction by transforming it into a difference. When you have a fraction

• For instance, given a logarithmic function such as \( \log_b \frac{4}{5} \), you can apply the quotient rule.
• According to the rule: \[\log_b \left( \frac{4}{5} \right) = \log_b 4 - \log_b 5\]
• This transformation turns the fraction into a subtraction of two logarithms.
• Why is this useful? This allows you to make use of known logarithm values, instead of calculating the log of a fraction from scratch.

Understanding the quotient rule helps in tackling a wide range of mathematical problems where fractions are involved.
Remember that the base of the logarithms must remain the same when applying this rule. This maintenance ensures the log transformation is consistent and accurate.

Base Change in Logarithms

Sometimes you need to change the base of a logarithm to make it easier to work with. This is particularly useful when you have a log with a base that isn’t easily calculable or when comparing logs of different bases. Here's how the process works:

• If you have \( \log_b a \), you can change it to another base \( c \) using:\[\log_b a = \frac{\log_c a}{\log_c b}\]
• This means you divide the log of the number by the log of the original base, both calculated to your new base \( c \).
• A common choice for \( c \) is 10 (common logarithm) or \( e \) (natural logarithm), but you can choose any base that simplifies the problem.
• This change of base formula is particularly handy when using calculators, as many only include functions for base 10 or \( e \) logarithms.

Understanding and using base change allows for greater flexibility in calculation and problem-solving, enhancing your mathematical toolkit.

Decimal Approximation of Logarithms

Logarithms can often result in lengthy decimals when calculated exactly. Therefore, decimal approximation is a practice used to simplify these numbers, making them more workable.

• To perform a decimal approximation, you typically round the result to a certain number of decimal places.
• In the case of the problem given, the result \(-0.0969\) was rounded to four decimal places.
• Rounding ensures precision in a way that fits the problem needs, while keeping calculations manageable.

Decimal approximation is fundamental in both theoretical and practical applications of mathematics.
Using approximations strategically is crucial: adjust the decimal length to balance between accuracy and simplicity for your specific task. This skill is invaluable in making complex math more accessible and easier to interpret.