Definition 6.3.2. Logarithm.
\(\log_a(x)=y\) if and only if \(a^y=x\)
Section 6.3 Logarithm Properties
Consider the two graphs in Figure 6.3.1. Note they are the same data drawn with different scales. Write down coordinates for the first three points. Is it easier to be precise using the version on the left or the version on the right?
The graph on the left uses the typical (linear) scale. Specifically on both the x and y axes each tick represents the same difference (add 1 unit for each tick or 2 units for each label). The graph on the right is the same data using a logarithmic scale. The x axis is still linear (same as the left), but the y axes is labeled such that each tick is the same product (multiply by 10 for each label).
Logarithms were developed initially to manage data that is very large. They are effective for data that grows quickly. Measurement of acidity (pH) and volume (decibel) both use a logarithmic scale. Studies of the human brain indicate that our brains interpret sensory data using a logarithmic scale. For example, a 10 degree temperature change feels like more of change if the initial temperature is \(23^\circ\) F than if it is \(51^\circ\) F.
Logarithms also have a connection to drunk pigeons and a Scotsman’s bones.
Subsection 6.3.1 Definition of Logarithm
The following definition is commonly used for algebraic uses of logarithms. Note that logarithms are defined here as the opposite of an exponential.
Example 6.3.3.
Each of the following is a conversion between exponential and logarithmic notation.
(a)
\(\log_2(8)=3\) is the same as \(2^3=8\text{.}\)
(b)
\(\log_5(1/25)=-2\) is the same as \(5^{-2}=1/25\text{.}\)
(c)
\(\log_{10}(\sqrt{10})=1/2\) is the same as \(10^{1/2}=\sqrt{10}\text{.}\)
Checkpoint 6.3.4.
Write \(\log_2(16)=4\) in exponential notation. Confirm that it is true.
Solution.
Using Definition 6.3.2 we obtain \(2^4=16\text{.}\) By multiplying we confirm it is true.
Checkpoint 6.3.5.
Write \(100^{3/2}=1000\) in logarithm notation.
Solution.
Using Definition 6.3.2 we obtain \(\log_{100}(1000)=3/2\text{.}\)
Subsection 6.3.2 Graphing Logarithms
Now we will use Definition 6.3.2 to analyze the shape of graphs of logarithms.
To practice we will graph \(y=\log_2(x)\text{.}\) As before we will begin by completing a table. First, consider \(y=\log_2(2)\text{.}\) This can be re-written as \(2^y=2\) which tells us that \(y=1\text{.}\) Second, consider \(y=\log_2(4)\text{.}\) This can be re-written as \(2^y=4\) which tells us that \(y=2\text{.}\) Because we re-write as \(2^y=x\) it is easiest if the \(x\) is a power of 2. For example \(x=2^3=8\text{.}\) This tells us that \(\log_2(8)=3\text{.}\)
Note that this last point shows us that the logarithm grows (vertically) forever. If we want the height to be 100, we select \(x=2^{100}\text{.}\)
This also shows us that the rate of growth is very slow. To increase one unit of height the input must double (because this is base 2).
Consider next the special case \(y=\log_2(1)\text{.}\) This can be re-written as \(2^y=1\) which tells us that \(y=0\text{.}\) This is the only place where a logarithm is zero.
Next we will plot points left of \(x=1\text{.}\) We start with fractions. For \(y=\log_2(1/2)\) we have \(2^y=\frac{1}{2}\text{.}\) This tells us that \(y=-1\) because negative exponents mean division. Similarly for \(y=\log_2(1/4)\) we have \(2^y=\frac{1}{4}=\frac{1}{2^2}\text{,}\)so \(y=-2\text{.}\)
Just as the curve grows, slowly forever as \(x\) increases, these examples show us that it grows forever downward as \(x\) approaches zero. Notice the y value doubled when we cut the input in half. This means the growth downward is very fast.
| \(x\) | \(\log_2(x)\) | |
| 1 | \(\log_2(1)=\) | 0 |
| 2 | \(\log_2(2)=\) | 1 |
| 4 | \(\log_2(4)=\) | 2 |
| 8 | \(\log_2(8)=\) | 3 |
| \(2^{10}\) | \(\log_2(2^{10})=\) | 10 |
| 1/2 | \(\log_2(1/2)=\) | -1 |
| 1/4 | \(\log_2(1/4)=\) | -2 |
There are two types of values we did not include in our graph example.
Example 6.3.8.
What is \(y=\log_2(0)\text{?}\)
Solution.
First we re-write this log as an exponential. \(2^y=0\text{.}\) We know that no exponent will produce a zero, so there is no solution for this.
Checkpoint 6.3.9.
Re-write \(\log_2(-4)=x\) as an exponential. Can you find a number for \(x\) that makes this statement true?
Subsection 6.3.3 Special Logarithms
Some logarithms occur sufficiently frequently in applications that they have their own notation.
Logarithms were developed to manage large numbers base 10. Thus we call base 10 logarithms: common logs. This is written without the base. For example \(\log(100)=2\) is the same as \(\log_{10}(100)=2\text{.}\)
In science and from mathematics we need another log called the natural logarithm. This is written as \(\ln(x)\) (for logarithm natural). Natural logarithms are paired with the base \(e\text{.}\) So \(\ln(5)=y\) is the same as \(e^y=5\text{.}\) \(e\) is a naturally occuring constant. You do not need to memorize an approximation (your calculator can handle that for you). For the curious \(e \approx 2.1718281828\text{.}\)
Checkpoint 6.3.10.
Using technology, calculate \(\ln({8})=\) .
Precision should be at least four decimal places.
Answer.
\(2.0794\)
Exercises 6.3.4 Exercises
Exercise Group.
Calculate using logarithms
1. Notation.
2. Notation.
3. Notation.
4. Notation.
5. Notation.
6. Evaluate.
7. Evaluate.
8. Notation.
9. Notation.
10. Solve.
11. Solve.
12. Solve.
13. Solve.
14. Solve.
15. Solve.
16. Solve.
17. Application.
Exercise Group.
Graph Logarithms
