Logarithmic function properties, examples, exercises

The logarithmic function is a mathematical relationship that associates each positive real number x with its logarithm Y on a base to. This relation meets the requirements to be a function: each element x belonging to the domain has a unique image.

Therefore:

f (x) = y = log_to x , with a> 0 and different from 1.

The main properties of the logarithmic function are:

-Its domain is all reals greater than 0, not including 0. In other words, there is no logarithm of 0 or negative numbers in any base. In interval form:

Sun _F = (0, ∞ +)

-The logarithm of a number can be negative, positive or 0, so its range or range is:

Rgo _F = (-∞, ∞ +)

-The logarithmic function is always increasing for a> 1 and decreasing for a<1.

-The inverse of f (x) = log_to x is the exponential function.

In effect, the logarithm function based on, is the inverse function of the potential function:

F^-1(x) = a^Y

Since the logarithm in base to of a number x, It's the number Y to which the base should be raised to to get x.

-The logarithm of the base is always 1. Thus, the graph of f (x) = log_to x always intersects the x-axis at the point (1,0)

-The logarithmic function is transcendent and it cannot be expressed as a polynomial or as a quotient of these. In addition to the logarithm, this group includes the trigonometric and exponential functions, among others.

Article index

• 1 Examples
• 2 The advantages of logarithms
  • 2.1 Example
• 3 Application exercise: the Richter scale
  • 3.1 Solution
• 4 References

Examples

The logarithmic function can be established by various bases, but the most used are 10 and and, where and is the Euler number equal to 2.71828 ... .

When the base 10 is used the logarithm is called a decimal logarithm, vulgar logarithm, Briggs or just plain logarithm.

And if the number e is used, then it is called a natural logarithm, by John Napier, the Scottish mathematician who discovered logarithms..

The notation used for each one is the following:

-Decimal logarithm: log₁₀ x = log x

-Natural logarithm: ln x

When another base is to be used, 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_two 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_two 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 above values ​​are true:

10¹ = 10

and^2,3026 = 10,0001

two^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..

Furthermore, logarithms have the following properties:

-Product: log (a.b) = 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..

Let's see an example of the handling of the properties of logarithms:

Example

Find the value of x in the following expression:

log (5x +1) = 1 + log (2x-1)

Answer

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 the 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 [(5x + 1) / (2x-1)] = 1

However, on the right is the number 1, which we can express as log 10, as we saw earlier. Then:

log [(5x + 1) / (2x-1)] = log 10

For equality to be fulfilled, 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 it is given by the formula:

M_R = log (10⁴ 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 (10⁴ 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 = 10⁴ I

The intensity of the earthquake in Mexico was:

I_M = 10 ^7.7 / 10⁴ = 10^3.7

 -chili: 9.5 = log (10⁴ I)

The same procedure takes us to the intensity of the Chilean earthquake I_Ch:

I_Ch = 10 ^9.5 / 10⁴ = 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

1. Carena, M. 2019. Pre-University Mathematics Manual. National University of the Litoral.
2. Figuera, J. 2000. Mathematics 1st. Diversified Year. CO-BO editions.
3. Jiménez, R. 2008. Algebra. Prentice hall.
4. Larson, R. 2010. Calculation of a variable. 9th. Edition. Mcgraw hill.
5. Stewart, J. 2006. Precalculus: Mathematics for Calculus. 5th. Edition. Cengage Learning.