Problem 35 - Calculus for Biology and Medicine

Question

Find the following numbers on a number line that is on a logarithmic scale (base 10\(): 10^{2}, 10^{-3}, 10^{-4}, 10^{-7}\), and \(10^{-10}\). (b) Can you find 0 on a number line that is on a logarithmic scale? (c) Can you find negative numbers on a number line that is on a logarithmic scale?

Step-by-Step Solution

Answer

Logarithmic plots: 100 at 2, 0.001 at -3, 0.0001 at -4, etc. 0 and negatives can't be represented.

Step 1: Understanding the Logarithmic Scale

A logarithmic scale on a number line plots numbers according to their powers of 10 rather than their actual values. If you compare two points on this line, the distance corresponds to the logarithm (base 10) of their ratio.

Step 2: Plotting 10^2 on the Number Line

The number \(10^2\) is equal to 100. On a logarithmic scale, this number is represented at the point where the exponent 2 is plotted, or 2 units to the right of 1 (the reference point for \(10^0\)).

Step 3: Plotting 10^{-3} on the Number Line

The number \(10^{-3}\) is equal to 0.001. On a logarithmic scale, it is represented at the point \(-3\), as we plot the exponent on the scale.

Step 4: Plotting 10^{-4} on the Number Line

The number \(10^{-4}\) is equal to 0.0001. It is plotted at the point \(-4\) on the logarithmic scale.

Step 5: Plotting 10^{-7} on the Number Line

The number \(10^{-7}\) equals 0.0000001. Therefore, it is plotted at \(-7\) on the logarithmic scale.

Step 6: Plotting 10^{-10} on the Number Line

The number \(10^{-10}\) equals 0.0000000001, which is plotted at \(-10\) on the logarithmic scale.

Step 7: Can Zero Be Plotted on a Logarithmic Scale?

Zero cannot be represented on a logarithmic scale because you cannot take the logarithm of zero; specifically, \(\log_{10}(0)\) is undefined. This is because there is no power of 10 that will result in 0.

Step 8: Can Negative Numbers Be Plotted on a Logarithmic Scale?

Negative numbers cannot be plotted on a standard logarithmic scale. This is because the logarithm of a negative number is not defined in the real numbers; \(\log_{10}(-x)\) is undefined.

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics that describe how many times a number, known as the base, is multiplied by itself. The exponent is the small number written slightly above and to the right of a base number. For example, in the expression \(10^2\), the base is 10 and the exponent is 2. This means that 10 is multiplied by itself once to get 100.
The result is also known as a power of 10. Exponents can be positive, zero, or negative.

• Positive exponents, like 2 in \(10^2\), indicate that the base is multiplied by itself.
• Zero exponents, like in \(10^0\), mean that any nonzero base raised to zero equals 1.
• Negative exponents, such as -3 in \(10^{-3}\), signify that the base is divided by itself, which results in fractions like 0.001.

Understanding exponents helps in navigating a logarithmic scale, where numbers are plotted according to their exponents, not their actual sizes.

Number Line

A number line is a visual representation of numbers in an ordered, straight line. On a typical number line, numbers increase as you move to the right and decrease as you move to the left. However, a logarithmic number line is quite different.

On a logarithmic scale, numbers are plotted based on the exponent they are raised to when the base is 10. Instead of evenly spaced intervals, the spacing between points on the line reflects the logarithms of the values.

• \(10^2\) or 100 is plotted at 2, because on a logarithmic scale, we are interested in the exponent, which in this case, is 2.
• For fractions like \(10^{-4}\), which equals 0.0001, the point is plotted at -4.

This unique property makes logarithmic scales incredibly useful for dealing with data that spans many orders of magnitude. However, you'll notice that zero and negative numbers have their complications on this type of number line, as they cannot be represented under our usual rules.

Logarithm

A logarithm is the inverse operation to exponentiation and provides a way to determine how many times one number must be multiplied by itself to achieve another number. When you're using base 10, you often see this referred to as "log" or \(\log_{10}\).

The logarithm answers the question: "To what power must 10 be raised to get a particular number?" For instance, \(\log_{10}(100) = 2\), because 10 raised to the power of 2 is 100. This property is the backbone of plotting numbers on a logarithmic scale.

• Positive numbers find a representation, like \(10^2\) where \(\log_{10}(100) = 2\).
• Zero cannot be represented since \(\log_{10}(0)\) is undefined—it’s impossible for any power of 10 to result in 0.
• Negative numbers also pose a challenge because their logs are not defined in real numbers without extending into complex numbers.

Logarithmic scales, while confusing at first, become powerful tools in scientific and engineering fields. They help us manage extensive data ranges more simply, but require a good understanding of both exponents and logarithms to be fully comfortable.