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Section2.7Properties of Logarithms

Subsection2.7.1Overview

We previously learned that logarithms are the inverse functions of exponential functions. In this section, we will further develop our understanding of logarithms by considering a logarithmic number line. This new number line has the property that what we normally think of as addition is interpreted in the context of multiplication, resulting from the fact that when exponentials are multiplied, the powers are added. This interpretation allows us to understand the basic properties of logarithms.

In addition, we will learn that when exponential and power functions can be plotted with logarithmic scale axes, the resulting transformed graph will appear linear. Not only will this allow us to identify from data whether a power function or exponential function is appropriate, but by finding the appropriate equation of a line (in the transformed variables), we can find the equation of the actual function using the properties of the logarithm.

Subsection2.7.2Logarithmic Number Lines

The standard number line is also called an arithmetic number line. Spacing between numbers is based on addition. That is, there is a length called the unit length that represents the space between consecutive whole numbers, and in particular it is the distance between the numbers 0 and 1 on the number line. A consequence of this spacing of numbers is that the space between any two numbers \(a\) and \(b\) on the number line is exactly the same as the spacing between \(0\) and \(b-a\).

Adding numbers on an arithmetic number line corresponds geometrically to combining the lengths. One way to see this is to consider two identical number lines and two numbers \(a\) and \(b\). Take the first number line and find the number \(a\) on that number line. Now, take the second number line and move it so that the number 0 (which is the additive identity) is exactly aligned with the number \(a\) on the first number line. Find the second number \(b\) on the shifted number line and see what number appears next to it on the original number line. That number will be \(a+b\).

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Figure2.7.1 Addition of the numbers 8 and -3 using arithmetic number lines ends at \(8+-3=5\).

A logarithmic number line is a number line that is designed for multiplication instead of addition. The number 1, which is the multiplicative identity, represents the origin. The logarithmic number line involves only positive numbers and is designed so that the spacing between values corresponds to the ratio of numbers (instead of the difference). That is, given two positive numbers \(a>0\) and \(b>0\), the spacing between the numbers is the same as the space between 1 and \(b/a\).

The consequence of this construction is that geometric addition corresponds to multiplication of numbers. That is, if we want to multiply two numbers \(a\) and \(b\), you start with one logarithmic number line and position the second line so 1 is aligned with \(a\). Finding the number \(b\) on the second line will correspond to the location of \(ab\) on the original line.

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Figure2.7.2 Multiplication of the numbers 3 and 5 using logarithmic number lines ends at \(3 \cdot 5=15\).

This number line exactly corresponds to the idea of the logarithm functions. Consider an exponential base \(b\). The integer powers of \(b\) will be equally spaced on the logarithmic number line. The unit length will correspond to the spacing between the origin 1 and the base \(b\) which is also the spacing between all integer powers of \(b\). Adjacent to the logarithmic number line, put an arithmetic number line so that the origin 0 is adjacent to the 1 on the logarithmic line and the number 1 is adjacent to the base \(b\) on the logarithmic line. The exponential function with base \(b\), \(\exp_b(x)\), is the mapping from the arithmetic line to the logarithmic line. The logarithm function with base \(b\), \(\log_b(x)\), is the inverse mapping from the logarithmic line back to the arithmetic line. The following figure illustrates this with a base \(b=3\).

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Figure2.7.3 The logarithm base \(b=3\) uses the logarithmic scale with a unit length corresponding to \(b=3\).
Definition2.7.4
For a base \(b>0\) with \(b \ne 1\), we define \(\log_b(x)\) to be that number \(a\) such that \(x=\exp_b(a)=b^a\). \begin{equation*} \log_b(x) = a \quad \Leftrightarrow \quad \exp_b(a)=x \quad \Leftrightarrow \quad b^a = x \end{equation*}

Because multiplication on the logarithmic scale is identical to addition on the arithmetic scale, the logarithm function inherits a corresponding property. Also, division is the inverse operation to multiplication in the same way that subtraction is the inverse operation to addition. Finally, because powers correspond to repeated multiplication, any power \(a^p\) in a logarithmic scale will correspond to a position that is exactly \(p\) times the logarithm of \(a\). These three features provide the three key properties of every logarithm function, which are summarized in the following theorem.

These properties are exactly the inverse properties of exponential functions.

Proof

The logarithm properties can be used to find logarithms of a quantity in terms of its factors. This can be illustrated using numbers as well as formulas.

Example2.7.7
Given that \(\log_3(2) \approx 0.6309\) and \(\log_3(5) \approx 1.4650\), determine each of the following:
  1. \(\log_3(10)\)
  2. \(\log_3(\frac{27}{5})\)
  3. \(\log_3(\frac{32}{125})\)
Solution
Example2.7.8

Expand the expression \(\displaystyle \log_b(\frac{u\cdot v^k}{w^n})\) using properties of logarithms.

Solution

The reverse process can also be performed, taking an expression involving multiple logarithms that are added or subtracted to find a single expression involving one logarithm. The key to doing this properly is to be sure that order of operations actually gives you a sum of two logarithms (to become the logarithm of a product), a difference of two logarithms (to become the logarithm of a quotient), or a product of a value times a logarithm (to become the logarithm of a power).

Example2.7.9

Simplify the expression \(\log(x) - 4\log(y) + 2 \log(z)\) using properties of logarithms.

Solution

Scientific calculators have two logarithm functions. One logarithm that is used most often for data involves base \(b=10\), also called the common logarithm. This is especially useful as it relates to scientific notation. Every positive number can be written in scientific notation as \(m \times 10^p\) where \(1 \le m \lt 10\) and \(p\) is an integer. Then the common logarithm will be \begin{equation*} \log_{10}(m \times 10^p) = p + \log_{10}(m) \end{equation*} with a value between \(p\) and \(p+1\).

Example2.7.10
\begin{align*} \log_{10}(3.25) & \approx 0.5119 \\ \log_{10}(3.25 \times 10^4) & = 4 + \log_{10}(3.25) \approx 4.5119 \\ \log_{10}(3.25 \times 10^{-3}) & = -3 + \log_{10}(3.25) \approx -2.4881 \end{align*}

While a common logarithm is convenient for representing numbers in terms of their order of magnitude, mathematics usually considers the natural logarithm with base \(e \approx 2.7182818285\). The reason for this choice will be made clear as we learn calculus.

Subsection2.7.3Semi–Log and Log–Log Plots

When data grow multiplicatively, such as in exponential growth, a logarithmic scale is more appropriate than a standard arithmetic (linear) scale. Instead of plotting the data in a standard Cartesian plane, one or both axes are replaced with a logarithmic number line. A log–log plot has both axes using a logarithmic scale. A semi-log plot has only one axis changed to a logarithmic scale.

In addition to helping visualize data that have a wide range of order of magnitudes (the power in scientific notation), logarithmic scales provide a useful transformation. In particular, power functions and exponential functions transform into linear functions under appropriate logarithmic transformations. This section explains how this can be used.

An elementary power function is a function where the independent variable is raised to a constant power \(p \ne 0\), usually a rational number. \begin{equation*} y = \mathrm{pow}_{p}(x) = x^p \end{equation*} Every other power function is a constant multiple of an elementary power function, \begin{equation*} y = A \cdot x^p, \end{equation*} where \(A\) is some constant value.

An elementary exponential function is a function where a constant base \(b>0\) with \(b \ne 1\) is raised to power given by the independent variable. \begin{equation*} y = \exp_{b}(x) = b^x \end{equation*} Every other exponential function is a constant multiple of an elementary exponential function, \begin{equation*} y = A \cdot b^{x}, \end{equation*} where \(A\) is a constant.

Be sure that you see the difference between a power function, where the base is the variable and the power is a constant, and an exponential function where the base is a constant and the power is the variable.

Now, consider what happens if you use a logarithm to transform the dependent variable in each case. This corresponds to making the \(y\)-axis a logarithmic scale.

Suppose that \(y\) is a power function with \(A>0\): \begin{equation*} y = A \cdot x^{p}. \end{equation*} Under a transformation, we have \(v = \log(y)\) so that the power function becomes: \begin{equation*} v = \log(y) = \log(A \cdot x^{p}) = \log(A) + \log(x^{p}) = \log(A) + p \log(x) \end{equation*} Notice that the independent variable is also in a logarithm, so we make the \(x\)-axis a logarithmic scale corresponding to a transformation \(u = \log(x)\). In the transformed variables, we have \begin{equation*} v = \log(A) + p \cdot u. \end{equation*} That is, a power function in linear in a log–log graph. The slope of the line corresponds to the power \(p\).

Suppose that \(y\) is an exponential function with \(A>0\) and base \(b>0\): \begin{equation*} y = A \cdot b^{x}. \end{equation*} Under a transformation, we have \(v = \log(y)\) so that the exponential function becomes: \begin{equation*} v = \log(y) = \log(A \cdot b^x) = \log(A) + \log(b^x) = \log(A) + x \log(b) \end{equation*} This time, the independent variable is not in a logarithm, so the \(x\)-axis is kept in a standard axis. An exponential function appears linear in a semi-log graph. The slope of the line is the logarithm of the base.

Example2.7.11
Find the power function passing through the points \((1,5)\) and \((4,30)\).
Solution
Example2.7.12
Find the exponential function passing through the points \((1,5)\) and \((4,30)\).
Solution

The previous examples illustrated how to find approximate solutions. Whenever you replace an exact expression with a rounded decimal value, you are doing an approximation. The same problems can be solved exactly by using properties of logarithms.

Example2.7.13
Find the exact power function passing through the points \((1,5)\) and \((4,30)\).
Solution
Example2.7.14
Find the exact exponential function passing through the points \((1,5)\) and \((4,30)\).
Solution