- Examples
- The advantages of logarithms
- Example
- Reply
- Application exercise: the Richter scale
- Solution
- References
The logarithmic function is a mathematical relationship that associates each positive real number x with its logarithm y on a base a. This relation meets the requirements to be a function: each element x belonging to the domain has a unique image.
Thus:
Since the logarithm based on a number x is the number y to which the base a must be raised to obtain x.
-The logarithm of the base is always 1. Thus, the graph of f (x) = log a x always intersects the x-axis at the point (1,0)
-The logarithmic function is transcendent and cannot be expressed as a polynomial or as a quotient of these. In addition to the logarithm, this group includes the trigonometric functions and the exponential, among others.
Examples
The logarithmic function can be established by various bases, but the most used are 10 and e, where e is the Euler number equal to 2.71828….
When base 10 is used, the logarithm is called a decimal logarithm, ordinary logarithm, Briggs' or just plain logarithm.
And if the number e is used, then it is called a natural logarithm, after John Napier, the Scottish mathematician who discovered logarithms.
The notation used for each one is the following:
-Decimal logarithm: log 10 x = log x
-Neperian logarithm: ln x
When you are going to use another base, it is absolutely necessary to indicate it as a subscript, because the logarithm of each number is different depending on the base to be used. For example, if it is logarithms in base 2, write:
y = log 2 x
Let's look at the logarithm of the number 10 in three different bases, to illustrate this point:
log 10 = 1
ln 10 = 2.30259
log 2 10 = 3.32193
Common calculators only bring decimal logarithms (log function) and natural logarithm (ln function). On the Internet there are calculators with other bases. In any case, the reader can verify, with its help, that the previous values are satisfied:
10 1 = 10
e 2.3026 = 10.0001
2 3.32193 = 10.0000
Small decimal differences are due to the number of decimal places taken in calculating the logarithm.
The advantages of logarithms
Among the advantages of using logarithms is the ease they provide to work with large numbers, using their logarithm instead of the number directly.
This is possible because the logarithm function grows more slowly as the numbers get larger, as we can see in the graph.
So even with very large numbers, their logarithms are much smaller, and manipulating small numbers is always easier.
In addition, logarithms have the following properties:
- Product: log (ab) = log a + log b
- Quotient: log (a / b) = log a - log b
- Power: log a b = b.log a
And in this way, the products and quotients become additions and subtractions of smaller numbers, while the potentiation becomes a simple product even though the power is high.
That is why logarithms allow us to express numbers that vary in very large ranges of values, such as the intensity of sound, the pH of a solution, the brightness of stars, the electrical resistance and the intensity of earthquakes on the Richter scale.
Figure 2. Logarithms are used on the Richter scale to quantify the magnitude of earthquakes. The image shows a collapsed building in Concepción, Chile, during the 2010 earthquake. Source: Wikimedia Commons.
Let's see an example of the handling of the properties of logarithms:
Example
Find the value of x in the following expression:
Reply
We have here a logarithmic equation, since the unknown is in the argument of the logarithm. It is solved by leaving a single logarithm on each side of the equality.
We start by placing all terms that contain "x" to the left of the equality, and those that contain only numbers to the right:
log (5x + 1) - log (2x-1) = 1
On the left we have the subtraction of two logarithms, which can be written as the logarithm of a quotient:
log = 1
However, on the right is the number 1, which we can express as log 10, as we saw earlier. So:
log = log 10
For equality to be true, the arguments of the logarithms must be equal:
(5x + 1) / (2x-1) = 10
5x + 1 = 10 (2x - 1)
5x + 1 = 20 x - 10
-15 x = -11
x = 11/15
Application exercise: the Richter scale
In 1957 an earthquake occurred in Mexico whose magnitude was 7.7 on the Richter scale. In 1960 another earthquake of greater magnitude occurred in Chile, of 9.5.
Calculate how many times the earthquake in Chile was more intense than the one in Mexico, knowing that the magnitude M R on the Richter scale is given by the formula:
M R = log (10 4 I)
Solution
The magnitude on the Richter scale of an earthquake is a logarithmic function. We are going to calculate the intensity of each earthquake, since we have the Richter magnitudes. Let's do it step by step:
- Mexico: 7.7 = log (10 4 I)
Since the inverse of the logarithm function is the exponential, we apply this to both sides of the equality with the intention of solving for I, which is found in the argument of the logarithm.
Since they are decimal logarithms, the base is 10. Then:
10 7.7 = 10 4 I
The intensity of the Mexico earthquake was:
I M = 10 7.7 / 10 4 = 10 3.7
- Chile: 9.5 = log (10 4 I)
The same procedure leads us to the intensity of the Chilean I Ch earthquake:
I Ch = 10 9.5 / 10 4 = 10 5.5
Now we can compare both intensities:
I Ch / I M = 10 5.5 / 10 3.7 = 10 1.8 = 63.1
I Ch = 63.1. I M
The earthquake in Chile was about 63 times more intense than the one in Mexico. Since the magnitude is logarithmic, it grows more slowly than the intensity, so a difference of 1 in the magnitude, means a 10 times greater amplitude of the seismic wave.
The difference between the magnitudes of both earthquakes is 1.8, therefore we could expect a difference in intensities closer to 100 than to 10, as it actually happened.
In fact, if the difference had been exactly 2, the Chilean earthquake would have been 100 times more intense than the Mexican one.
References
- Carena, M. 2019. Pre-University Mathematics Manual. National University of the Litoral.
- Figuera, J. 2000. Mathematics 1st. Diversified Year. CO-BO editions.
- Jiménez, R. 2008. Algebra. Prentice Hall.
- Larson, R. 2010. Calculation of a variable. 9th. Edition. McGraw Hill.
- Stewart, J. 2006. Precalculus: Mathematics for Calculus. 5th. Edition. Cengage Learning.