Problem 14 - Algebra 2

Question

Use a calculator to evaluate each expression to four decimal places. $$ \log 5 $$

Step-by-Step Solution

Answer

The approximate value of \( \log 5 \) to four decimal places is 0.6990.

Step 1: Set Up the Problem

We are asked to evaluate the logarithm of 5, denoted as \( \log 5 \). Logarithms with no indicated base typically use base 10, which is known as the common logarithm.

Step 2: Use the Calculator

Enter \( \log(5) \) into a scientific calculator. Make sure your calculator is set to "log" for base 10. If unsure, consult your calculator’s manual for instructions.

Step 3: Round to Four Decimal Places

After inputting the operation, your calculator will display a decimal number. It should show approximately 0.69897. Round this number to four decimal places, resulting in 0.6990.

Key Concepts

Common Logarithm

A logarithm is a mathematical operation that helps find the exponent needed to raise a base to a certain number. When we use "log" without a specified base, we assume it's base 10. This is specifically called a **common logarithm**.

• Common logarithms are used primarily in the sciences and engineering because they simplify complex calculations.
• The notation "\( \log 5 \)" indicates a common logarithm, where you need to find the power to which the base 10 is raised to get the number 5.

An easy way to think about it is: if you know 10 is to be raised to some power \( x \), such that \( 10^x = 5 \), this power \( x \) is the value of "\( \log 5 \)".
Common logarithms simplify calculations because they can transform multiplicative relationships into additive ones. This is useful for computations involving large numbers.

Scientific Calculator

A **scientific calculator** is a special type of calculator designed to handle a range of mathematical operations, including logarithms. Here’s how you can make use of it for calculating common logarithms:

• Make sure your calculator is set to interpret logarithms as base 10. This is the default setting on most scientific calculators.
• To calculate \( \log 5 \), simply input "log" followed by the number "5" and press enter or "=" depending on your model.
• If unaware of your calculator's functionality, it's a good idea to look at the user manual for guidance.

Scientific calculators make it possible to quickly compute the values of logarithms, besides being used for other advanced calculations like trigonometry and exponential functions. They are an essential tool for navigating high school math and scientific courses.

Rounding Decimals

**Rounding decimals** is a critical skill in mathematics to ensure that calculations are easy to understand and close to the intended accuracy. When you calculate a logarithm using a scientific calculator, you often get a result with many decimal places.

• We usually round the result to a certain number of decimal places. In this exercise, we are to round to four decimal places.
• Look at the fifth decimal to decide. If it's 5 or more, round the fourth decimal up. If it's less than 5, the fourth decimal stays the same.

For instance, with "\( \log 5 \)", the calculator gives us about 0.69897. Since the fifth decimal "7" is greater than 5, we round up, leaving us with 0.6990. This rounding ensures precision while keeping the number manageable and easy to work with.