Problem 47 - 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. $$ \log \left(1.3 \times 10^{7}\right) $$

Step-by-Step Solution

Answer

The answer will be the output of the floor function that is applied to the real number value of the logarithm.

Step 1: Evaluate the logarithm

Input the value \( \log \left(1.3 \times 10^{7}\right) \) to the calculator to find the real number.

Step 2: Apply Floor Function

After obtaining the real number, apply the floor function to it to get the largest integer that is less than the calculated decimal value. The floor function essentially rounds down a number to the nearest, less, whole number.

Key Concepts

Calculator Use for Logarithms

Using a calculator to evaluate logarithms can make complex problems much easier. When you input the logarithm into a calculator, it processes the equation and provides the result to several decimal places.
In the example, we calculate \( \log \left(1.3 \times 10^{7}\right) \). To input this:

• Key in the logarithm function \( \log \).
• Enter the number \( 1.3 \times 10^{7} \), using the exponent key for powers of ten.
• Press "equals" or "enter" for the result.

Your calculator should display the logarithm to the set decimal places, often four. Accurate input is essential for a precise output, so be meticulous when entering values.
If your calculator has a decimal setting, ensure it's set correctly to showcase four decimal places for consistency.

Evaluating Logarithms

Evaluating a logarithm involves finding the exponent to which a fixed number, the base, must be raised to produce a particular number. In our exercise, we work with the logarithmic expression \( \log \left(1.3 \times 10^{7}\right) \). With base 10 logarithms, which are common, the equation asks: "To what power must\( 10 \) be raised to yield \( 1.3 \times 10^{7} \)?"
After entering this expression into your calculator, you receive a decimal answer. This result precisely shows how many times you'd multiply the base (10) by itself to achieve the initial number (1.3 multiplied by 10 to the 7th power).
For understanding:

• The decimal response is the exponent.
• The process simplifies finding complex powers, modernizing mathematical calculations.

Understanding the Floor Function

The floor function is a mathematical concept that rounds a real number down to the nearest whole number. It's crucial when you need to find the largest integer less than or equal to a given number. After evaluating a logarithm with a calculator, you often end up with a number that isn't a whole integer.
For instance, if you calculate a log and get 6.1131, the floor function translates this to 6. The reduction excludes all decimal portions.
Here's how the floor function works:

• Identify the numeric result of the logarithm.
• Remove the decimal, effectively finding the largest integer less than the number.
• Use it in contexts where whole numbers are necessary or preferred, such as counting items.

The floor function is a helpful tool, keeping computations straightforward and grounded, particularly when decimals aren't practical.