Problem 33 - Calculus for Biology and Medicine

Question

Find the following numbers on a number line that is on a logarithmic scale (base 10\(): 0.0002,0.02,1,5,50,100,1000,8000\), and 20000 .

Step-by-Step Solution

Answer

Convert each number to logarithmic scale values and plot them accordingly.

Step 1: Understand the Logarithmic Scale

A logarithmic scale means each step on the number line represents a power of 10. Thus, when placing numbers, we use the logarithm (base 10) of each number to determine positioning.

Step 2: Calculate the Logarithms

For each number, calculate its logarithm base 10: - For 0.0002: \( \log_{10} (0.0002) = -3.7 \)- For 0.02: \( \log_{10} (0.02) = -1.7 \)- For 1: \( \log_{10} (1) = 0 \)- For 5: \( \log_{10} (5) \approx 0.7 \)- For 50: \( \log_{10} (50) \approx 1.7 \)- For 100: \( \log_{10} (100) = 2 \)- For 1000: \( \log_{10} (1000) = 3 \)- For 8000: \( \log_{10} (8000) \approx 3.9 \)- For 20000: \( \log_{10} (20000) \approx 4.3 \)

Step 3: Plot the Logarithms on the Number Line

Place each number on the number line according to its calculated logarithmic value: - 0.0002 should be placed near -3.7 - 0.02 should be placed near -1.7 - 1 should be placed at 0 - 5 should be placed at approximately 0.7 - 50 should be at approximately 1.7 - 100 should be at 2 - 1000 should be at 3 - 8000 should be at approximately 3.9 - 20000 should be at approximately 4.3

Step 4: Verify and Adjust as Necessary

Ensure the numbers are correctly placed relative to one another. As numbers increase by a factor of 10, they should be equidistant along the number line on the logarithmic scale.

Key Concepts

Understanding Base 10 Logarithms

A base 10 logarithm is a mathematical way to express numbers using powers of 10. It helps in working with very large or small numbers in a simplified manner. Instead of dealing directly with these numbers, a logarithm allows us to handle their powers easily.
For example, the base 10 logarithm of 100 is 2, because 100 is 10 raised to the power of 2. Similarly, for a number like 0.01, the base 10 logarithm is -2, because 0.01 is equal to 10 raised to the power of -2.
Logarithms can turn multiplication into addition and division into subtraction, making calculations easier and more intuitive when working with extremely different scales.
Understanding this concept is crucial when using a logarithmic scale.

Navigating the Number Line

A number line is a visual representation of numbers plotted in a straight line. Normally, this line is marked with intervals evenly distributed by units. However, when a number line is set on a logarithmic scale, each step represents a power of 10 rather than units.
This means if we go from 1 to 10, 10 to 100, or 0.1 to 1, each movement signifies multiplying or dividing by 10.
Key Points for Logarithmic Number Lines:

• Equidistant steps indicate multiplication by a constant factor.
• Each step corresponds with a logarithmic increase or decrease.
• Perfect for representing data that varies exponentially.

By understanding the number line on a logarithmic scale, positioning and plotting becomes much more manageable.

Mastering Calculating Logarithms

Calculating logarithms is about determining to what power the base (10 in this case) must be raised to result in a given number. When you calculate the base 10 logarithm of a number, you're essentially asking "10 to what power equals this number?"
Here's how to calculate simple base 10 logarithms:

• Identify the number whose logarithm you need to find.
• Rewrite the number as a power of 10 (if possible).
• Express the number in terms of its power of 10; this power is your logarithm.

Practicing with numbers like 1, 10, 100, or 0.01 helps solidify this concept. Using calculators can simplify finding logarithms of less familiar numbers, such as 5 or 8000.

Plotting Numbers on a Logarithmic Scale

Plotting numbers on a logarithmic scale requires identifying and calculating their respective logarithms beforehand. Instead of using regular intervals, each plotted position reflects a power of 10 from the original number. Steps to plotting numbers:

• Calculate the base 10 logarithm of each number.
• Mark the corresponding points based on the calculated logarithms on the line.
• Ensure equidistance for every tenfold increase or decrease.

An important thing to note is that the scale helps reveal patterns and relationships in data that might not appear in linear scales. For example, exponential growth trends appear as straight lines, making analysis easier. This concept is handy when dealing with phenomena like population growth or chemical reactions that change vastly across a scale.