Question

Evaluate the logarithm using a calculator. Round your answer to three decimal places. \(\log \frac{2}{7}\)

Short answer

After using the calculator and rounding the result, the value of \( \log \frac{2}{7} \) to three decimal places is -0.565.

Step-by-step solution

Calculate the Logarithm

Using a scientific calculator or a computing software, compute the logarithm of the ratio between 2 and 7 (\( \log \frac{2}{7} \)).

Round the Result

Round the result obtained to three decimal places.

Key concepts

Scientific Calculator

A scientific calculator is an essential tool when working with logarithms, especially those that are not easily calculable by hand. A scientific calculator is capable of performing operations involving exponents, roots, and, importantly, logarithms.
It usually comes with a dedicated "log" button that allows you to compute base-10 logarithms directly. In the given exercise

• To compute the logarithm of a ratio such as \( \log \frac{2}{7} \), you would first need to enter the division: 2 divided by 7.
• After obtaining the decimal value of this division, you will press the "log" button to find its logarithm.
• The scientific calculator then displays the logarithmic value, which can be used for further rounding or calculation.

Understanding how to use a scientific calculator efficiently can save you time and improve your accuracy when dealing with complex mathematical functions like logarithms.

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify numbers while maintaining their general proximity to the original value.
This is particularly important when dealing with results from a scientific calculator, which often produces answers with many decimal places. In the logarithm exercise given, you are required to round your answer to three decimal places.Here's a quick guide to rounding:

• Look at the digit in the fourth decimal place – this is the critical digit.
• If it is greater than or equal to 5, increase the third decimal place by 1.
• If it is less than 5, keep the third decimal place as it is.
• After making this adjustment, all numbers after the third decimal become zero and are no longer shown.

For example, if your scientific calculator displays a result of \(-0.557231\), rounding to three decimal places results in \(-0.557\) since the fourth digit, 2, is less than 5. Remember, rounding helps to improve readability and usability of numerical data without significantly affecting accuracy.

Ratio

Ratios are a way of comparing two quantities by division. In the original exercise, the quantity \( \frac{2}{7} \) is a ratio representing a part-to-whole relationship. Ratios can be expressed in different forms, such as fractions, decimals, or percentages.
In this particular example, the ratio \( \frac{2}{7} \) represents the quantity of 2 compared to 7.

• Converting this ratio into a decimal by dividing 2 by 7 equals approximately 0.285714.
• This decimal is then used in calculating the logarithm with a scientific calculator.

Understanding ratios is crucial as they are widely used in topics involving proportions, speeds, and quantities, making them foundational in many fields of study.