Problem 42 - Algebra 2

Question

Use your calculator to evaluate each logarithm to four decimal places. Then find the largest integer that is less than the value of the logarithm. $$ \log \left(6.1 \times 10^{-5}\right) $$

Step-by-Step Solution

Answer

The value of the logarithm to four decimal places is -4.2144. The largest integer that is less than -4.2144 is -5.

Step 1: Evaluate The Logarithmic Expression

On your calculator, enter the expression \( \log \left(6.1 \times 10^{-5}\right) \). Make sure that the result is correct up to four decimal places.

Step 2: Determine The Largest Integer Less Than The Logarithmic Value

Now that you have the decimal value, examine which integer number is less than this value. This step may not need the use of a calculator as it mainly involves understanding the set of integer numbers.

Key Concepts

Logarithmic Expressions

Logarithmic expressions represent exponents in another form. When we talk about \( \log(a) \), we're finding an exponent that shows how many times we multiply a base number, which is usually 10 when indicated as "log," to get the number \( a \). For example, \( \log(100) = 2 \) because 10 raised to the second power is 100. Logarithms make it easier to work with very large or very small numbers by simplifying multiplication and division into addition and subtraction.

Understanding how to manipulate these expressions is crucial in various fields such as science and engineering. To grasp this concept fully, always remember that logs are just exponents seen from a different perspective. This understanding is essential when working through exercises that require finding the exact decimal representation of a logarithm.

• Logarithmic expressions simplify calculations.
• They convert multiplication into addition.
• Commonly used base is 10 ("log").

Evaluating Logarithms

When evaluating a logarithm like \( \log(6.1 \times 10^{-5}) \), the goal is to find its decimal equivalent using a calculator. Begin by inputting the expression correctly – ensure that you're using the base 10 log function on your device. Most calculators will provide you with a result that should be rounded or truncated to four decimal places for a clear and concise answer.

Break the problem down into straightforward steps:

• First, input the expression: enter \( 6.1 \times 10^{-5} \) directly into the calculator.
• Use the "log" function to compute the logarithm.
• Read the result on your calculator, often a negative number because you’re working with a small decimal value.
• Finally, round the result to four decimal places as required, ensuring accuracy.

Calculators make evaluating logarithms easy by automatically applying the underlying mathematical rules, giving you quick access to the exponents that represent complex numbers.

Integer Approximation

After evaluating a logarithm, another task often follows: finding the largest integer less than the result, which means locating the greatest whole number that is still smaller than the logarithm's decimal value. This technique is known as "integer approximation," and it helps simplify the interpretation of a result when an exact whole number is easier or more practical to use.

For example, if your calculator gives you \( -4.2143 \) as a result, the largest integer less than this value is \( -5 \), because \( -5 \) is smaller than \( -4.2143 \) even though it's numerically greater in absolute value. Recognizing that negative numbers work differently than positive ones is key here. This can sometimes trip people up, since it involves an understanding of how the number line and integer values operate.

Steps to approximate:

• Identify the decimal result from the logarithm.
• On direct inspection, determine which integer is closest but smaller.
• Check the placement of negative and positive numbers to ensure accuracy.

Mastering integer approximation teaches critical thinking about numerical values and their relationships.