Problem 88 - Intermediate Algebra

Question

Use a calculator to find each value. Give answers to four decimal places. See Using Your Calculator: Evaluating Logarithms. $$ \log 0.57 $$

Step-by-Step Solution

Answer

\( \log 0.57 \approx -0.2441 \)

Step 1: Understanding Logarithms

The logarithm of a number is the power to which the base (usually 10 for common logarithms) must be raised in order to obtain that number. Here, we need to find the common logarithm of 0.57, denoted as \( \log 0.57 \).

Step 2: Use a Scientific Calculator

To find \( \log 0.57 \), enter 0.57 into your scientific calculator and press the 'log' button. Modern calculators are designed to handle such calculations efficiently.

Step 3: Interpret Calculator Display

Once you press the 'log' button, the calculator will display the logarithmic value. Ensure your calculator is set to display the answer in decimal form and to four decimal places.

Step 4: Record the Answer

The calculator will provide an answer. Round this value to four decimal places as required by the problem.

Key Concepts

Scientific Calculator Use

Scientific calculators are essential tools in mathematics, especially for complex calculations like evaluating logarithms. These calculators have dedicated keys for functions such as logarithms, trigonometry, and exponentials. To compute a logarithm, simply input the number and press the 'log' key. Modern scientific calculators are designed for ease of use, typically displaying results instantly. They are portable devices, often powered by batteries, making them convenient for students and professionals alike. A key part of effectively using a scientific calculator is understanding its settings. Calculators need to be set to decimal mode to provide precise results. You can usually adjust the number of decimal places displayed in the calculator settings, which is useful for exercises requiring specific precision.

Evaluating Logarithms

The process of evaluating logarithms involves determining the power needed to raise a base (typically 10 for common logarithms) to get a specific number. For example, to find \( \log 0.57 \), we need to identify what power of 10 gives us 0.57.On a calculator, evaluating logarithms is straightforward. Enter the value for which you need the logarithm, and press the 'log' button. The calculator will display the result almost immediately.Understanding logarithms is crucial for various applications in science, technology, and finance. They help in managing exponential growth, calculating compounding interest, and understanding scientific phenoma, to name a few.

Decimal Places Rounding

Rounding to a specified number of decimal places is often necessary for precision in mathematical calculations. When a problem asks for answers to four decimal places, it is asking for accuracy up to 0.0001. Here's how to round correctly:

• Identify the digit at the fifth decimal place.
• If this digit is 5 or greater, increase the fourth decimal place by one.
• If it's less than 5, leave the fourth decimal place as is.

This practice ensures that your answer is both precise and consistent with mathematical standards. Precise rounding is particularly important in fields such as engineering and physics, where small differences can affect outcomes.