Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 5,620 $$

Step-by-Step Solution

Answer

The common logarithm of 5,620 is approximately 3.75.

Step 1: Understanding the Problem

We need to find the common logarithm (logarithm to the base 10) of the number 5,620 and round it to the nearest hundredth.

Step 2: Calculate the Common Logarithm

Use a calculator to find the common logarithm of 5,620. Input the number into the calculator and press the 'log' function. This gives us the value of \( \log_{10}(5620) \).

Step 3: Approximate the Result

The result from the calculator is approximately 3.74967. We need to round this to the nearest hundredth.

Step 4: Round to the Nearest Hundredth

To round 3.74967 to the nearest hundredth, look at the third decimal place (the thousandths place). Since it's more than 5, increase the second decimal place by one. Thus, 3.74967 rounds to 3.75.

Key Concepts

Logarithm Base 10

A common logarithm is simply a logarithm that has a base of 10. It's very handy for simplifying calculations with large numbers. The notation generally used is \( \log_{10} \) or just \( \log \) if the base is understood to be 10.

• The expression \( \log_{10}(x) \) asks: "To what power must 10 be raised, to equal \( x \)?"
• For example, \( \log_{10}(100) = 2 \) because \( 10^2 = 100 \).

Calculating the common logarithm of a number like 5,620 involves discovering the exponent that makes 10 raised to that power equal to 5,620. Using a calculator can help to quickly find results for complex numbers.

Rounding Decimals

Rounding decimals can be made simple with consistent practice. When rounding, you often focus on a particular decimal place. For example, rounding to the nearest hundredth means you look at the thousandths digit to decide.

• If the digit in the thousandths place is 5 or more, you round up the hundredths digit.
• If it's less than 5, you simply leave the hundredths digit as it is.

Take the number 3.74967: the digit 9 is in the hundredths place. The digit 6 is in the thousandths place, which is more than 5. Hence, you round the 9 up to 10. This effectively increases the digit in the tenths place by 1 due to carrying over, making the rounded number 3.75. Remember, rounding makes numbers easier to work with while retaining as much of their true value as necessary.

Using a Calculator

Using a calculator efficiently is a great skill, especially when working with functions like common logarithms. Modern calculators have specific buttons for logarithms, commonly labeled as 'log'.

Here's a simple guide:

• To find the logarithm of a number like 5,620, enter the number into the calculator.
• Press the 'log' button to see the result.

Most scientific calculators will handle this for you without extra input, providing you instantaneous results. For the common logarithm of 5,620, a calculator will directly give you a long decimal, but it helps you in rounding it to the desired decimal place for precision.