6.3 Logarithmic Functions
Figure 1 Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)
In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings, like those shown in Figure 1. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale whereas the Japanese earthquake registered a 9.0.
The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is \(10^{8 - 4} = 10^{4} = 10{,}000\) times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.
6.3.1 Converting from Logarithmic to Exponential Form
In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is \(10^{x} = 500,\) where \(x\) represents the difference in magnitudes on the Richter Scale. How would we solve for \(x?\)
We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve \(10^{x} = 500.\) We know that \(10^{2} = 100\) and \(10^{3} = 1000,\) so it is clear that \(x\) must be some value between 2 and 3, since \(y = 10^{x}\) is increasing. We can examine a graph, as in Figure 2, to better estimate the solution.
Figure 2
Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in Figure 2 passes the horizontal line test. The exponential function \(y = b^{x}\) is one-to-one, so its inverse, \(x = b^{y}\) is also a function. As is the case with all inverse functions, we simply interchange \(x\) and \(y\) and solve for \(y\) to find the inverse function. To represent \(y\) as a function of \(x,\) we use a logarithmic function of the form \(y = \log_{b}(x).\) The base \(b\) logarithm of a number is the exponent by which we must raise \(b\) to get that number.
We read a logarithmic expression as, “The logarithm with base \(b\) of \(x\) is equal to \(y,\) ” or, simplified, “log base \(b\) of \(x\) is \(y.\) ” We can also say, “ \(b\) raised to the power of \(y\) is \(x,\) ” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since \(2^{5} = 32,\) we can write \(\log_{2}32 = 5.\) We read this as “log base 2 of 32 is 5.”
We can express the relationship between logarithmic form and its corresponding exponential form as follows:
\[\log_{b}(x) = y\Leftrightarrow b^{y} = x,b > 0,b \neq 1\]
Note that the base \(b\) is always positive.
Because logarithm is a function, it is most correctly written as \(\log_{b}(x),\) using parentheses to denote function evaluation, just as we would with \(f(x).\) However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as \(\log_{b}x.\) Note that many calculators require parentheses around the \(x.\)
We can illustrate the notation of logarithms as follows:
Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means \(y = \log_{b}(x)\) and \(y = b^{x}\) are inverse functions.
6.3.2 Converting from Exponential to Logarithmic Form
To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base \(b,\) exponent \(x,\) and output \(y.\) Then we write \(x = \log_{b}(y).\)
6.3.3 Evaluating Logarithms
Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider \(\log_{2}8.\) We ask, “To what exponent must \(2\) be raised in order to get 8?” Because we already know \(2^{3} = 8,\) it follows that \(\log_{2}8 = 3.\)
Now consider solving \(\log_{7}49\) and \(\log_{3}27\) mentally.
- We ask, “To what exponent must 7 be raised in order to get 49?” We know \(7^{2} = 49.\) Therefore, \(\log_{7}49 = 2\)
- We ask, “To what exponent must 3 be raised in order to get 27?” We know \(3^{3} = 27.\) Therefore, \(\log_{3}27 = 3\)
Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate \(\log_{\frac{2}{3}}\frac{4}{9}\) mentally.
- We ask, “To what exponent must \(\frac{2}{3}\) be raised in order to get \(\frac{4}{9}?\) ” We know \(2^{2} = 4\) and \(3^{2} = 9,\) so \(\left( \frac{2}{3} \right)^{2} = \frac{4}{9}.\) Therefore, \(\log_{\frac{2}{3}}\left( \frac{4}{9} \right) = 2.\)
6.3.4 Using Common Logarithms
Sometimes you may see a logarithm written without a base. When you see one written this way, you need to look at the expression before evaluating it. It may be that the base you use doesn’t matter. If you find it in computer science, it often means \(\log_{2}(x)\). However, in mathematics it almost always means the common logarithm of 10. In other words, the expression \(\log(x)\) often means \(\log_{10}(x).\)
Currently, we use \({\log_{b}(x)},{\lg(x)}\) as the common logarithm, \({lb}(x)\) as the binary logarithm, and \(\ln(x)\) as the natural logarithm. Writing \(\lg(x)\) without specifying a base is now considered bad form, despite being frequently found in older materials.
6.3.5 Using Natural Logarithms
The most frequently used base for logarithms is \(e,\) the value of which is approximately \(2.71828\). Base \(e\) logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base \(e\) logarithm, \(\log_{e}(x),\) has its own notation, \(\ln(x).\)
Most values of \(\ln(x)\) can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, \(\ln 1 = 0.\) For other natural logarithms, we can use the \(\ln\) key that can be found on most scientific calculators. We can also find the natural logarithm of any power of \(e\) using the inverse property of logarithms.
Section Exercises
Verbal
1. What is a base \(b\) logarithm? Discuss the meaning by interpreting each part of the equivalent equations \(b^{y} = x\) and \(\log_{b}x = y\) for \(b > 0,b \neq 1.\)
Solution (click to reveal)
A logarithm is an exponent. Specifically, it is the exponent to which a base \(b\) is raised to produce a given value. In the expressions given, the base \(b\) has the same value. The exponent, \(y,\) in the expression \(b^{y}\) can also be written as the logarithm, \(\log_{b}x,\) and the value of \(x\) is the result of raising \(b\) to the power of \(y.\)
2. How is the logarithmic function \(f(x) = \log_{b}x\) related to the exponential function \(g(x) = b^{x}?\) What is the result of composing these two functions?
3. How can the logarithmic equation \(\log_{b}x = y\) be solved for \(x\) using the properties of exponents?
Solution (click to reveal)
Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation \(b^{y} = x,\) and then properties of exponents can be applied to solve for \(x.\)
4. Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base \(b,\) and how does the notation differ?
5. Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base \(b,\) and how does the notation differ?
Solution (click to reveal)
The natural logarithm is a special case of the logarithm with base \(b\) in that the natural log always has base \(e.\) Rather than notating the natural logarithm as \(\log_{e}(x),\) the notation used is \(\ln(x).\)
Algebraic
For the following exercises, rewrite each equation in exponential form.
6. \(\text{log}_{4}(q) = m\)
7. \(\text{log}_{a}(b) = c\)
Solution (click to reveal)
\(a^{c} = b\)
8. \(\log_{16}(y) = x\)
9. \(\log_{x}(64) = y\)
Solution (click to reveal)
\(x^{y} = 64\)
10. \(\log_{y}(x) = -11\)
11. \(\log_{15}(a) = b\)
Solution (click to reveal)
\(15^{b} = a\)
12. \(\log_{y}(137) = x\)
13. \(\log_{13}(142) = a\)
Solution (click to reveal)
\(13^{a} = 142\)
14. \(\text{log}(v) = t\)
15. \(\text{ln}(w) = n\)
Solution (click to reveal)
\(e^{n} = w\)
For the following exercises, rewrite each equation in logarithmic form.
16. \(4^{x} = y\)
17. \(c^{d} = k\)
Solution (click to reveal)
\(\text{log}_{c}(k) = d\)
18. \(m^{- 7} = n\)
19. \(19^{x} = y\)
Solution (click to reveal)
\(\log_{19}y = x\)
20. \(x^{-\frac{10}{13}} = y\)
21. \(n^{4} = 103\)
Solution (click to reveal)
\(\log_{n}(103) = 4\)
22. \(\left( \frac{7}{5} \right)^{m} = n\)
23. \(y^{x} = \frac{39}{100}\)
Solution (click to reveal)
\(\log_{y}\left( \frac{39}{100} \right) = x\)
24. \(10^{a} = b\)
25. \(e^{k} = h\)
Solution (click to reveal)
\(\text{ln}(h) = k\)
For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form.
26. \(\text{log}_{3}(x) = 2\)
27. \(\text{log}_{2}(x) = - 3\)
Solution (click to reveal)
\(x = 2^{- 3} = \frac{1}{8}\)
28. \(\text{log}_{5}(x) = 2\)
29. \(\log_{3}(x) = 3\)
Solution (click to reveal)
\(x = 3^{3} = 27\)
30. \(\text{log}_{2}(x) = 6\)
31. \(\text{log}_{9}(x) = \frac{1}{2}\)
Solution (click to reveal)
\(x = 9^{\frac{1}{2}} = 3\)
32. \(\text{log}_{18}(x) = 2\)
33. \(\log_{6}(x) = - 3\)
Solution (click to reveal)
\(x = 6^{- 3} = \frac{1}{216}\)
34. \(\text{log}(x) = 3\)
35. \(\text{ln}(x) = 2\)
Solution (click to reveal)
\(x = e^{2}\)
For the following exercises, use the definition of common and natural logarithms to simplify.
36. \(\text{log}(100^{8})\)
37. \(10^{\text{log}(32)}\)
Solution (click to reveal)
\(32\)
38. \(2\text{log}(.0001)\)
39. \(e^{\ln{(1.06)}}\)
Solution (click to reveal)
\(1.06\)
40. \(\ln\left( e^{- 5.03} \right)\)
41. \(e^{\ln{(10.125)}} + 4\)
Solution (click to reveal)
\(14.125\)
Numeric
For the following exercises, evaluate the base \(b\) logarithmic expression without using a calculator.
42. \(\text{log}_{3}\left( \frac{1}{27} \right)\)
43. \(\text{log}_{6}(\sqrt{6})\)
Solution (click to reveal)
\(\frac{1}{2}\)
44. \(\text{log}_{2}\left( \frac{1}{8} \right) + 4\)
45. \(6\text{log}_{8}(4)\)
Solution (click to reveal)
\(4\)
For the following exercises, evaluate the common logarithmic expression without using a calculator.
46. \(\text{log}(10{,}000)\)
47. \(\text{log}(0.001)\)
Solution (click to reveal)
\(- \text{3}\)
48. \(\text{log}(1) + 7\)
49. \(2\text{log}(100^{- 3})\)
Solution (click to reveal)
\(- 12\)
For the following exercises, evaluate the natural logarithmic expression without using a calculator.
50. \(\text{ln}(e^{\frac{1}{3}})\)
51. \(\text{ln}(1)\)
Solution (click to reveal)
\(0\)
52. \(\text{ln}(e^{- 0.225}) - 3\)
53. \(25\text{ln}(e^{\frac{2}{5}})\)
Solution (click to reveal)
\(10\)
Technology
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.
54. \(\text{log}(0.04)\)
55. \(\text{ln}(15)\)
Solution (click to reveal)
\(\text{2}.\text{7}0\text{8}\)
56. \(\text{ln}\left( \frac{4}{5} \right)\)
57. \(\text{log}(\sqrt{2})\)
Solution (click to reveal)
\(0.151\)
58. \(\text{ln}(\sqrt{2})\)
Extensions
59. Is \(x = 0\) in the domain of the function \(f(x) = \log(x)?\) If so, what is the value of the function when \(x = 0?\) Verify the result.
Solution (click to reveal)
No, the function has no defined value for \(x = 0.\) To verify, suppose \(x = 0\) is in the domain of the function \(f(x) = \log(x).\) Then there is some number \(n\) such that \(n = \log(0).\) Rewriting as an exponential equation gives: \(10^{n} = 0,\) which is impossible since no such real number \(n\) exists. Therefore, \(x = 0\) is not the domain of the function \(f(x) = \log(x).\)
60. Is \(f(x) = 0\) in the range of the function \(f(x) = \log(x)?\) If so, for what value of \(x?\) Verify the result.
61. Is there a number \(x\) such that \(\ln x = 2?\) If so, what is that number? Verify the result.
Solution (click to reveal)
Yes. Suppose there exists a real number \(x\) such that \(\ln x = 2.\) Rewriting as an exponential equation gives \(x = e^{2},\) which is a real number. To verify, let \(x = e^{2}.\) Then, by definition, \(\ln(x) = \ln\left( e^{2} \right) = 2.\)
62. Is the following true: \(\frac{\log_{3}(27)}{\log_{4}\left( \frac{1}{64} \right)} = -1?\) Verify the result.
63. Is the following true: \(\frac{\ln\left( e^{1.725} \right)}{\ln(1)} = 1.725?\) Verify the result.
Solution (click to reveal)
No; \(\ln(1) = 0,\) so \(\frac{\ln\left( e^{1.725} \right)}{\ln(1)}\) is undefined.
Real-World Applications
64. The exposure index \(EI\) for a camera is a measurement of the amount of light that hits the image receptor. It is determined by the equation \(EI = \log_{2}\left( \frac{f^{2}}{t} \right),\) where \(f\) is the “f-stop” setting on the camera, and \(t\) is the exposure time in seconds. Suppose the f-stop setting is \(8\) and the desired exposure time is \(2\) seconds. What will the resulting exposure index be?
65. Refer to the previous exercise. Suppose the light meter on a camera indicates an \(EI\) of \(- 2,\) and the desired exposure time is 16 seconds. What should the f-stop setting be?
Solution (click to reveal)
\(2\)
66. The intensity levels \(I\) of two earthquakes measured on a seismograph can be compared by the formula \(\log\frac{I_{1}}{I_{2}} = M_{1} - M_{2}\) where \(M\) is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0. How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.