Problem 49 - Precalculus : Building Concepts and Connections

Question

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$. \log _{7} 150$$

Step-by-Step Solution

Answer

After evaluating using a calculator and rounding off to four decimal places, the final answer to the problem is obtained.

Step 1: Apply the change-of-base formula

We begin by applying the change-of-base formula. This allows us to simplify the expression \(\log_7 150\) to a form we can easily calculate: \(\log_7 150 = \frac{\log 150}{\log 7}\). This formula allows us to calculate the original logarithm by taking any logarithm to the base of our choice for the numerator and denominator.

Step 2: Evaluate using a calculator

Next, the expression \(\frac{\log 150}{\log 7}\) is evaluated using a calculator. Note that the calculator may use either base 10 (common logarithm) or base e (natural logarithm) for the calculations. However, as long as the same base is used for both the numerator and the denominator, the result will be the same.

Step 3: Round the answer to four decimal places

Finally, the answer obtained from the calculator is rounded to four decimal places as requested in the question in order to give the final answer.

Key Concepts

Logarithm Evaluation

Understanding how to evaluate logarithms is crucial for various fields of math and science. A logarithm, essentially, tells us the power to which we must raise a certain number, called the base, to get another number. For instance, if we know that 7^x = 150, we are searching for the value of x, and this is expressed as \( \log_7 150 \).

The change-of-base formula is a powerful tool when dealing with bases that are not 10 or e (the natural logarithm base). The formula is \( \log_b a = \frac{\log a}{\log b} \), where a is the number we take the logarithm of, b is the base of the logarithm, and the logarithms on the right side of the equation can be with respect to any base. It effectively transforms the evaluation of logarithms with unusual bases into a task manageable by most calculators, which typically only handle base 10 or e.

Calculator Usage for Logarithms

Calculators become indispensable when dealing with logarithmic calculations that cannot be easily simplified by hand. To evaluate logarithms effectively using a calculator, there are a few steps to consider.

Most scientific calculators have a LOG button for common logarithms (base 10) and an LN button for natural logarithms (base e). When using the change-of-base formula, you enter the numerator, which is the LOG or LN of the number, followed by the denominator, which is the LOG or LN of the base.

Example Calculation

For the expression \( \log_7 150 \), you would input LOG (or LN) of 150 divided by LOG (or LN) of 7 into your calculator. Remember to confirm that your calculator is set to the correct mode (degrees or radians) as needed for your calculations, although this does not impact logarithms directly.

Logarithmic Expressions Rounding

The last step in logarithm evaluation is often to round the result to a specified number of decimal places, which is essential for reporting numerical answers and ensuring they are practical to use. The level of precision required for a problem is not always the same—sometimes two decimal places are enough, and other times four or more are necessary.

To round a logarithmic expression correctly:

• Identify the digit required for the final decimal place.
• Look at the next digit (to the right). If this digit is five or greater, round the last required digit up by one.
• If the next digit is less than five, leave the last required digit as it is.

Applying this to the provided example, we evaluate \( \frac{\log 150}{\log 7} \) and then round the answer to four decimal places. This precision ensures the answer is suitable for most applications while retaining a significant level of accuracy.