Problem 60 - Algebra for College Students

Question

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\log 2}{5 \log 1.02} $$

Step-by-Step Solution

Answer

The result is approximately 7.0 when rounded to the nearest hundredth.

Step 1: Understanding the Expression

The given expression is \( \frac{\log 2}{5 \log 1.02} \). Our task is to calculate this expression and round the final answer to the nearest hundredth.

Step 2: Calculate the Logarithms

First, calculate the logarithm \( \log 2 \) and \( \log 1.02 \) using a calculator. \( \log 2 \approx 0.3010 \) and \( \log 1.02 \approx 0.0086 \).

Step 3: Multiply the Denominator

Multiply the denominator by 5. Calculate \( 5 \times \log 1.02 = 5 \times 0.0086 \approx 0.0430 \).

Step 4: Compute the Division

Divide the result from Step 2 by the result from Step 3. Calculate \( \frac{0.3010}{0.0430} \approx 7.0 \).

Step 5: Round the Result

Round the result from the division to the nearest hundredth. In this case, the answer 7.0 is already rounded to the nearest hundredth.

Key Concepts

Scientific Calculator Use

Scientific calculators are powerful tools that can handle various mathematical operations beyond basic arithmetic. One common use is calculating logarithms, such as \( \log 2 \) or \( \log 1.02 \). To use a scientific calculator for logarithms:

• First, ensure your calculator is in the standard mode (not radians or degrees).
• Look for the "log" button, typically it calculates the common logarithm (base 10).
• Enter the number for which you want the logarithm and press the "log" button. For example, typing "2" followed by "log" should display approximately 0.3010.

Consistent practice will help you become quicker and more accurate when using a scientific calculator, making complex calculations like division of logarithms more manageable.

Rounding Numbers

Rounding numbers is a crucial skill in mathematics, especially when precision is important but not critical to the outcome. In the given exercise, we need to round to the nearest hundredth.

• The hundredth place is two digits to the right of the decimal point.
• Look at the digit in the thousandth place, immediately to the right of the hundredth place. If it is 5 or greater, round the hundredth place up by 1.
• If the thousandth place digit is less than 5, keep the hundredth place digit the same.

For instance, when you have 7.00 as the result, it's already at the nearest hundredth. But if you had 7.008, you would round it to 7.01 because the digit immediately following the hundredth place is greater than or equal to 5.

Division of Logarithms

Dividing logarithms can seem complicated at first, but it becomes simpler if you break it down step by step. In our given expression \( \frac{\log 2}{5 \log 1.02} \), here’s how it works:

• Calculate each logarithm individually using a calculator. In this case, \( \log 2 \approx 0.3010 \) and \( \log 1.02 \approx 0.0086 \).
• Multiply the logarithm in the denominator by 5 (since it’s \( 5 \log 1.02 \)), which yields \( 5 \times 0.0086 = 0.0430 \).
• Perform the division between the result of \( \log 2 \) and \( 5 \log 1.02 \), which results in \( \frac{0.3010}{0.0430} \approx 7.0 \).

Breaking down the process into these simple steps can help you manage and understand the division of logarithmic expressions efficiently.