Problem 45 - Algebra 2

Question

Use your calculator to evaluate each logarithm to four decimal places. Then find the largest integer that is less than the value of the logarithm. $$ \log \frac{1}{6} $$

Step-by-Step Solution

Answer

After evaluating \( \log \left(\frac{1}{6}\right) \) to four decimal places, find the largest integer by rounding down to the nearest whole number.

Step 1: Calculate the Logarithm

Use a scientific calculator to evaluate the logarithm \( \log \left(\frac{1}{6}\right) \). Make sure the calculator is set to the correct base (common logarithm base 10) if necessary.

Step 2: Determine the Decimal Value

After calculating, round the result to four decimal places as instructed.

Step 3: Find the Largest Integer

Identify the largest integer that is less than the calculated decimal value of the logarithm. If the value is negative, remember that the largest integer less than the value will be more negative.

Key Concepts

Scientific Calculator Usage

When faced with the need to calculate logarithms, a scientific calculator is an indispensable tool. To correctly utilize it, first ensure that the calculator is set to use the logarithm base you need. Commonly for logarithms without a specified base—the notation simply as \( \log \)—the base is assumed to be 10. On your calculator, this function is usually labeled 'LOG'.

Input the number or expression you wish to find the logarithm of, in this case \( \frac{1}{6} \), by pressing the LOG button followed by \( \frac{1}{6} \), or its decimal equivalent. The calculator will display the result, which you'll need to round as instructed. Using your calculator effectively requires familiarity with these functions and careful entry of the numbers to ensure accuracy.

Common Logarithms

Common logarithms are those that have a base of 10, often denoted as \( \log \) without an explicit base written. They are used widely in both mathematics and sciences due to their utility in describing exponential growth or decay. You computed \( \log \left(\frac{1}{6}\right) \) likely using a common logarithm. Here, the idea is to find the power to which the base 10 must be raised to obtain \( \frac{1}{6} \).

Understanding the relationship between exponents and logarithms is crucial, but in practice, for common logarithms, you will primarily use the 'LOG' key on a scientific calculator, simplifying the process considerably.

Rounding Decimals

Rounding decimals is an essential skill in mathematics, especially when dealing with non-integer values. The instructions here specify rounding to four decimal places, which means you need to look at the fifth decimal place to decide whether to round up or down. If the fifth decimal is 5 or more, you round the fourth decimal up; if it's less than 5, you leave it as it is.

For example, if your calculator shows 0.12345, you would round this to 0.1235. If it were 0.12344, it would round to 0.1234. Always ensure precision in your rounding as it affects the subsequent step – finding the correct integer value.

Integer Values

An integer value is a whole number that can be positive, negative, or zero, but it does not include any fraction or decimal. In problems asking for the largest integer 'less than' a given number, you're performing what's known as the 'floor' operation. This means if you have a decimal value from your logarithm calculation, for instance, -0.3010, the largest integer less than this is -1, since -1 is the next whole number below -0.3010 without going over it.

It's common to think of positive numbers in this context, but for negative results, always remember that the integer less than the negative decimal is the more negative one, not the one closer to zero.