Problem 55 - Intermediate Algebra

Question

For Problems \(54-61\), perform the following calculations and express answers to the nearest hundredth. (These calculations are in preparation for our work in the next section.) $$ \frac{\ln 2}{\ln 7} $$

Step-by-Step Solution

Answer

The answer is approximately 0.36.

Step 1: Recall the Formulas

To solve this problem, we need to remember the logarithmic rule for division: \( \frac{\ln a}{\ln b} \) means using the natural logarithms for values \(a=2\) and \(b=7\). We will use a calculator to find \( \ln 2 \) and \( \ln 7 \).

Step 2: Calculate Natural Logarithms

We calculate the natural logarithm of 2 and 7 using a calculator: \( \ln 2 \approx 0.6931 \) and \( \ln 7 \approx 1.9459 \). These values are approximately computed to four decimal places.

Step 3: Perform the Division

Now, divide the calculated logarithms: \( \frac{\ln 2}{\ln 7} = \frac{0.6931}{1.9459}\). We use a calculator to perform this division.

Step 4: Round to Nearest Hundredth

After performing the division, \( \frac{0.6931}{1.9459} \approx 0.3561\). Round this value to the nearest hundredth, which gives us approximately 0.36.

Key Concepts

Logarithmic Division

Logarithmic division is all about dividing the values of logarithms. In math, this operation helps simplify expressions involving logarithms of numbers. For our specific task, we have two logarithms: \( \ln 2 \) and \( \ln 7 \). These represent the natural logarithms of 2 and 7, respectively. To divide these two, you perform a straightforward division: \( \frac{\ln 2}{\ln 7} \).

Here's a quick recap on why logarithmic division is useful:

• It allows comparison of growth rates between different exponential functions.
• It simplifies complex equations involving logarithms.
• Helps in solving real-life problems involving exponential change, such as population growth or interest rates.

To solve \( \frac{\ln 2}{\ln 7} \), you first need to calculate each natural logarithm using a calculator, leading you to approximate values which you can then divide, as done in our exercise.

Mathematical Approximation

Mathematical approximation is the process of finding values that are close to the exact numbers. It's a crucial skill in math, especially when dealing with complex calculations like logarithms. Initially, we calculated \( \ln 2 \approx 0.6931 \) and \( \ln 7 \approx 1.9459 \). These values are not exact, but they are close enough for practical purposes.

We use approximations when:

• The exact value is difficult or impossible to find.
• We need to simplify calculations for quick answers.
• The scenario doesn't require high precision.

One key detail in approximation is deciding how many decimal places to use. In our case, we used four decimal places for precision but might decide differently based on context. It's an important decision, balancing between simplicity and precision.

Rounding Decimals

Rounding decimals simplifies numbers by reducing the number of decimal places, which can be helpful for making the results more relatable and easier to communicate. In our calculation, we found that \( \frac{0.6931}{1.9459} \approx 0.3561 \). To express this to the nearest hundredth, we check the hundredth place and the digit right after it.

Here's how rounding works:

• If the digit after your rounding place is 5 or more, round up.
• If the digit is less than 5, round down.

So for 0.3561, since the digit in the thousandths place (6) is more than 5, we round the hundredths place from 5 to 6, resulting in 0.36. This process is vital in math and real-life applications, making numbers more understandable while retaining their usefulness.