Problem 61 - Algebra for College Students

Question

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\log 5}{3 \log 1.07} $$

Step-by-Step Solution

Answer

\(7.93\)

Step 1: Identify the Problem

We need to calculate the expression \( \frac{\log 5}{3 \log 1.07} \) and express the answer to the nearest hundredth. This involves using logarithms and basic arithmetic.

Step 2: Calculate Individual Logarithms

First, calculate the values of the logarithms in the expression. Use a calculator to find: \( \log 5 \) and \( \log 1.07 \). Typically, \( \log 5 \approx 0.69897 \) and \( \log 1.07 \approx 0.02938 \).

Step 3: Calculate the Denominator

Multiply the logarithm of 1.07 by 3 to calculate the denominator of the fraction:\[ 3 \times \log 1.07 = 3 \times 0.02938 \approx 0.08814 \]

Step 4: Divide to Find the Result

Compute the division of the two results from the previous steps:\[ \frac{\log 5}{3 \log 1.07} = \frac{0.69897}{0.08814} \] Using a calculator: \( \approx 7.928 \).

Step 5: Round the Answer

Round the result from the previous step to the nearest hundredth:\[ 7.928 \rightarrow 7.93 \]

Key Concepts

Understanding Basic Arithmetic

Basic arithmetic involves fundamental mathematical operations including addition, subtraction, multiplication, and division. These operations allow us to carry out a range of calculations. Arithmetic forms the basics of most mathematical concepts, including logarithms.

Logarithms themselves are not purely arithmetic but rely on these operations for their calculation. When calculating logarithms, you often need to perform basic arithmetic with other numbers to solve expressions as we saw in the exercise. This includes summing or multiplying logarithmic results with numbers to obtain new values.

• For instance, multiplying a logarithm by a whole number, as seen in step 3, to compute the denominator of the fraction.
• Understanding these arithmetic basics helps simplify more complex equations by breaking them down into manageable parts.

The Basics of Division

Division is one of the four fundamental arithmetic operations where you split a number into equal parts. In mathematical terms, it helps find how many times one number is contained within another.

In the given exercise, after calculating the logarithmic values and computing the denominator, division was the key operation needed. It involves taking the result from numerator and dividing it by the result from the denominator.

• The expression \( \frac{\log{5}}{3\log{1.07}} \) required dividing the value \(0.69897\) by \(0.08814\), which results in approximately \(7.928\).
• Mastery in division ensures that complex expressions can be solved efficiently.
• This operation is crucial in converting the fraction into a decimal, which then can be rounded as needed.

Rounding Numbers for Precision

Rounding numbers is a mathematical technique used to simplify numbers, making them easier to work with while maintaining a level of precision. This often involves reducing the number of digits to the right of the decimal point.

In mathematics, especially in calculations involving decimals, rounding helps to present results that are easier to interpret, without significant loss of accuracy.

• In our exercise, we rounded the decimal \(7.928\) to the nearest hundredth, achieving a result of \(7.93\).
• The process entailed looking at the third digit after the decimal to determine if the second digit needs to increase by one, which is important in achieving correct rounding.
• Rounding becomes particularly useful in scientific calculations where extreme precision is not necessary but clarity is important.