Problem 3 - Intermediate Algebra

Question

For Problems \(1-10\), use a calculator to find each common logarithm. Express answers to four decimal places. \(\log 52.23\)

Step-by-Step Solution

Answer

\(\log 52.23 \approx 1.7177\)

Step 1: Understand the Problem

We need to calculate the common logarithm, denoted as \( \log \), of the number 52.23. The common logarithm has base 10.

Step 2: Use a Calculator

Turn on your calculator and find the logarithm function key, usually labeled as 'log.' Enter the number 52.23 to compute its logarithm.

Step 3: Enter the Number

Input the number 52.23 into your calculator. Ensure you are using the correct function corresponding to the base 10 logarithm.

Step 4: Calculate the Logarithm

Press the '=' button or the calculate button after entering 52.23. The calculator should display the result of \( \log(52.23) \).

Step 5: Round the Result

Round the result from the calculator to four decimal places. This is the final answer we need for the logarithm of 52.23 to four decimal places.

Key Concepts

Calculator Usage

Understanding how to use a calculator efficiently is crucial, especially when calculating common logarithms. To begin with, turn on your calculator. Most scientific calculators have a specific key labeled 'log' for computing logarithms with a base of 10. This key simplifies finding the logarithm of any positive number quickly. To calculate \( \log(52.23) \), enter the number '52.23' into your calculator. Ensure that you've selected the 'log' function before proceeding.

• Find and press the 'log' key usually located near trigonometric functions.
• Input the number, which in our case is 52.23.
• Press the calculate button, often marked as '=' or 'enter'.

Following these steps will provide you with the logarithm value you need. Correct calculator usage makes dealing with logarithmic problems much simpler, allowing you to confidently solve similar tasks.

Decimal Places

When solving exercises involving logarithms, it is vital to understand decimal places. Rounding to decimal places ensures precision and consistency in your results. For this specific task, you're required to round the result to four decimal places. Decimal places follow the decimal point and dictate the number of digits to leave intact for accuracy.In practice, here's how you manage decimal places:

• After computing \( \log(52.23) \) with your calculator, you'll receive a number like 1.717575735.
• To simplify this, look at the fifth digit after your intended decimal place. If it's 5 or more, round up. If it's less, keep your current digit the same.
• For example, 1.717575735 would round to 1.7176 because the fifth digit, '7', prompts you to round up the fourth digit.

Rounding to the correct decimal places keeps your work organized and helps when comparing logarithmic values accurately.

Logarithmic Functions

Logarithmic functions are an essential concept in mathematics that allow us to solve equations involving exponential growth or decay. A common logarithm, like the one in our example \( \log 52.23 \), has a base of 10, denoted as \( \log_{10} \). This type of function is the inverse of exponential functions with the same base.Here's why logarithmic functions matter:

• They convert multiplicative processes into additive ones, making complex calculations easier.
• They are widely used in science, engineering, and data analysis to deal with exponential changes.
• Understanding them allows for better grasp of how exponential equations can be rearranged and solved.

In practice, common logarithms simplify real-world calculations by providing a straightforward method to interpret orders of magnitude. Mastery of logarithmic functions not only aids in mathematical computation but enhances analytical skills necessary across various disciplines.