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leARnIng ObjeCTIveS
In this section, you will:
- Convert from logarithmic to exponential form.
- Convert from exponential to logarithmic form.
- Evaluate logarithms.
- Use common logarithms.
- Use natural logarithms.
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[19]. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,[20]^ 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[21]^ whereas the Japanese earthquake registered a 9.0.[22]
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 = 104 = 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.
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 10x^ = 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 10x^ = 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.
400 x
y = 500
y = 10 x
y
200
600
1 2
800
3
1,
Figure 2 19 http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013. 20 http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc001xgp/#summary. Accessed 3/4/2013. 21 http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013. 22 http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc001xgp/#details. Accessed 3/4/2013.
- Identify the domain of a logarithmic function.
- Graph logarithmic functions.
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 = bx^ 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 = logb(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: logb(x) = y ⇔ b y^ = x, b > 0, b ≠ 1 Note that the base b is always positive.
logb(x) = y Think b to the y = x
=
to Because logarithm is a function, it is most correctly written as logb(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:
logb(c) = a means b a^ = c
=
to Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means y = logb (x) and y = bx^ are inverse functions.
definition of the logarithmic function A logarithm base b of a positive number x satisfies the following definition. For x > 0, b > 0, b ≠ 1, y = logb(x) is equivalent to b y^ = x where,
- we read logb(x) as, “the logarithm with base b of x” or the “log base b of x.”
- the logarithm y is the exponent to which b must be raised to get x. Also, since the logarithmic and exponential functions switch the x and y values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,
- the domain of the logarithm function with base b is (0, ∞).
- the range of the logarithm function with base b is ( −∞, ∞).
Q & A… Can we take the logarithm of a negative number? No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.
How To… Given an equation in logarithmic form logb(x) = y, convert it to exponential form.
- Examine the equation y = logb(x) and identify b, y, and x.
- Rewrite logb(x) = y as b y^ = x.
Example 3 Solving Logarithms Mentally Solve y = log 4 (64) without using a calculator. Solution First we rewrite the logarithm in exponential form: 4y^ = 64. Next, we ask, “To what exponent must 4 be raised in order to get 64?” We know 4 3 = 64 therefore, log 4 (64) = 3.
Example 4 Evaluating the Logarithm of a Reciprocal
Evaluate y = log (^3) _^1 27 ^
without using a calculator.
Solution First we rewrite the logarithm in exponential form: 3 y^ = _^1 27
. Next, we ask, “To what exponent must 3 be raised in order to get _^1 27
We know 3^3 = 27, but what must we do to get the reciprocal, _^1 27
? Recall from working with exponents that b−a^ = _^1 b a^
We use this information to write 3 −^3 = __^1 33 = __^1 Therefore, log^27 3 ^
_^1
27 ^
Using Common logarithms
Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression log(x) means log 10 (x). We call a base-10 logarithm a common logarithm. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.
definition of the common logarithm A common logarithm is a logarithm with base 10. We write log 10 (x) simply as log(x). The common logarithm of a positive number x satisfies the following definition. For x > 0, y = log(x) is equivalent to 10y^ = x We read log(x) as, “the logarithm with base 10 of x” or “log base 10 of x.” The logarithm y is the exponent to which 10 must be raised to get x.
How To… Given a common logarithm of the form y = log(x), evaluate it mentally.
- Rewrite the argument x as a power of 10: 10 y^ = x.
- Use previous knowledge of powers of 10 to identify y by asking, “To what exponent must 10 be raised in order to get x?”
How To… Given a common logarithm with the form y = log(x), evaluate it using a calculator.
- Press [LOG].
- Enter the value given for x, followed by [ ) ].
- Press [ENTER].
Example 6 Finding the Value of a Common Logarithm Using a Calculator Evaluate y = log(321) to four decimal places using a calculator. Solution
- Press [LOG].
- Enter 321, followed by [ ) ].
- Press [ENTER]. Rounding to four decimal places, log(321) ≈ 2.5065.
Analysis Note that 102 = 100 and that 103 = 1000. Since 321 is between 100 and 1000 , we know that log(321) must be between log(100) and log(1000). This gives us the following: 100 < 321 < 1000 2 < 2.5065 < 3 Example 7 Rewriting and Solving a Real-World Exponential Model The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation 10x^ = 500 represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes? Solution We begin by rewriting the exponential equation in logarithmic form. 10 x^ = 500 log(500) = x Use the definition of the common log.
Next we evaluate the logarithm using a calculator:
- Press [LOG].
- Enter 500, followed by [ ) ].
- Press [ENTER].
- To the nearest thousandth, log(500) ≈ 2.699. The difference in magnitudes was about 2.699.
Example 5 Finding the Value of a Common Logarithm Mentally
Evaluate y = log(1,000) without using a calculator. Solution First we rewrite the logarithm in exponential form: 10y^ = 1,000. Next, we ask, “To what exponent must 10 be raised in order to get 1,000?” We know 10^3 = 1,000 therefore, log(1,000) = 3.
(^0500) 1,
2
4
6
8
10
12
14
16
18
20
1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000 5,500 6, Account balance
The balance reaches $5, near year 14
Years
Figure 3
Finding the domain of a logarithmic Function Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined. Recall that the exponential function is defined as y = b x^ for any real number x and constant b > 0, b ≠ 1, where
- The domain of y is (−∞, ∞).
- The range of y is (0, ∞). In the last section we learned that the logarithmic function y = logb(x) is the inverse of the exponential function y = bx. So, as inverse functions:
- The domain of y = logb(x) is the range of y = bx^ : (0, ∞).
- The range of y = logb(x) is the domain of y = bx^ : (−∞, ∞).
Graphs of Logarithms
In Graphs of Exponential Functions , we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect.
To illustrate, suppose we invest $2,500 in an account that offers an annual interest rate of 5%, compounded continuously.
We already know that the balance in our account for any year t can be found with the equation A = 2500 e0.05t.
But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? Figure 3 shows this point on the logarithmic graph.
Logarithmic Model Showing Years as a Function of the Balance in the Account
How To… Given a logarithmic function, identify the domain.
- Set up an inequality showing the argument greater than zero.
- Solve for x.
- Write the domain in interval notation.
Example 9 Identifying the Domain of a Logarithm What is the domain of f (x) = log 2 (x + 3)?
Solution The logarithmic function is defined only when the input is positive, so this function is defined when x + 3 > 0. Solving this inequality, x + 3 > 0 The input must be positive. x > − 3 Subtract 3. The domain of f (x) = log 2 (x + 3) is (−3, ∞).
Example 10 Identifying the Domain of a Logarithm What is the domain of f (x) = log(5 − 2 x)? Solution The logarithmic function is defined only when the input is positive, so this function is defined when 5 − 2 x > 0. Solving this inequality, 5 − 2 x > 0 − 2 x > − 5
x < __^52
The input must be positive. Subtract 5.
Divideby− 2 and switch the inequality.
The domain of f (x) = log(5 − 2 x) is 5 –^ ∞,^ _^ .
Graphing logarithmic Functions Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function y = logb(x) along with all its transformations: shifts, stretches, compressions, and reflections.
We begin with the parent function y = logb(x). Because every logarithmic function of this form is the inverse of an exponential function with the form y = b x, their graphs will be reflections of each other across the line y = x. To illustrate this, we can observe the relationship between the input and output values of y = 2 x^ and its equivalent x = log 2 (y) in Table 1.
x −^3 −^2 −^1 0 1 2
2 x^ = y
_^1
_^1
_^1 1 2 4
log 2 (y) = x − 3 − 2 − 1 0 1 2
Table 1
Using the inputs and outputs from Table 1 , we can build another table to observe the relationship between points on the graphs of the inverse functions f (x) = 2 x^ and g(x) = log 2 (x). See Table 2.
f (x) = 2 x −3,^1 8
_ ^ −2,
_ (^) −1,^1 2
_ (^) (0, 1) (1, 2) (2, 4) (3, 8)
g(x) = log 2 (x) (^) 8
_^1 , − (^3) 4
_^1 , − (^2) 2
_^1 , − (^1) (1, 0) (2, 1) (4, 2) (8, 3)
Table 2
Example 11 Graphing a Logarithmic Function with the Form f ( x) = log b ( x).
Graph f (x) = log 5 (x). State the domain, range, and asymptote.
Solution Before graphing, identify the behavior and key points for the graph.
- Since b = 5 is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote x = 0, and the right tail will increase slowly without bound.
- The x-intercept is (1, 0).
- The key point (5, 1) is on the graph.
- We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see Figure 6 ).
x
f (x)
- 10 – 8 – 6 – 4 – (^2) – 1
5 4 3 2 1 2 4 6 8 10 (1,(1,
(5, 1) (^) f (x) = log 5 (x)
x = 0
Figure 6 The domain is (0, ∞), the range is (−∞, ∞), and the vertical asymptote is x = 0.
