# Applications of Logarithmic Functions

There are several applications of logarithmic functions in everyday life. Logarithmic functions allow us to model certain real-life situations. For example, we can use logarithmic scales to measure earthquake intensities (Ritcher scale) and to create the decibel scale of sound and the pH scale. We can also obtain models for chemical regulatory dissolution and information entropy.

##### ALGEBRA

Relevant for

Learning about some of the applications of logarithmic functions.

See applications

##### ALGEBRA

Relevant for

Learning about some of the applications of logarithmic functions.

See applications

## Important Applications of Logarithmic Functions

Logarithmic functions have applications in several areas. The following are some of the most important:

### Algebra and Calculus

Logarithmic functions are used in calculus to solve problems involving exponential growth and decay. For example, they can be used to find the instantaneous rate of change of a function or to solve differential equations involving exponential functions.

### Physics

Logarithmic functions describe phenomena such as sound intensity and earthquakes. For example, the decibel scale, which is used to measure the intensity of sound, is based on a logarithmic function.

In seismology, the Richter scale, which measures the magnitude of earthquakes, is also based on a logarithmic function.

### Engineering

Logarithmic functions are used to design and analyze electronic circuits, control systems, and mechanical systems. For example, in electronic engineering, logarithmic functions are used to design and analyze amplifiers, filters, and oscillators.

In control systems, logarithmic functions are used to model system behavior and design controllers.

In mechanical engineering, logarithmic functions are used to analyze the behavior of structures and design mechanical systems.

### Computer Science

In computer science, logarithmic functions help in algorithm analysis and the design of efficient algorithms. An example is the analysis of the temporal and spatial complexity of algorithms.

Many algorithms, such as binary search and the quick sort algorithm, have logarithmic time complexity, making them very efficient for large data sets.

### Economy

Economic growth can be analyzed with logarithmic functions and this makes it possible to analyze market trends. For example, they are used to model GDP growth and the spread of inflation.

### Chemistry

Logarithmic functions can be used to describe the behavior of chemical reactions. For example, they are used to describe the rate of chemical reactions, which is often measured in terms of the change in concentration of a reactant or product over time.

### Astronomy

Logarithmic functions allow us to describe the brightness of stars and the distance of galaxies. For example, the magnitude scale, which is used to measure the brightness of stars, is based on a logarithmic function.

Logarithmic functions are also used to estimate the distance of galaxies, which can be very large.

### Meteorology

With the use of logarithmic functions, it is possible to describe the behavior of atmospheric pressure and temperature. For example, these functions are used to model the behavior of atmospheric pressure and temperature at different altitudes.

## Magnitude of an earthquake

One of the applications of logarithmic functions is the measurement of earthquake intensities (Ritcher scale), sound (decibels), and bases and acids (pH). Let’s analyze the measurement of earthquake intensities.

In 1935 Charles Ritcher defined the magnitude of an earthquake with the formula:

$latex M=\log (\frac{I}{S})$

where I is the intensity of the earthquake measured by the amplitude of a seismometer taken 100 km from the epicenter and S is the intensity of a standard earthquake, which is defined with an amplitude of 1 micrometer or $latex {{10}^{- 4}}$ cm.

This means that the magnitude of a standard earthquake is:

$latex M=\log (\frac{S}{S}) = \log(1) = 0$

One of the largest earthquakes on record had a magnitude of 8.9 on the Ritcher scale. This would be equivalent to an intensity of 800,000,000. This means that the Ritcher scale allows us to obtain more manageable numbers.

Each increase of a number on the Ritcher scale indicates a 10-fold increase in intensity. For example, an earthquake with a magnitude of 6 is ten times stronger than an earthquake with a magnitude of 5. An earthquake with a magnitude of 8 is 100 times stronger than an earthquake with a magnitude of 6

### EXAMPLE

At the beginning of the century, an earthquake in California registered 8.3 on the Ritcher scale. In the same year, another earthquake was recorded in South America that was 4 times stronger. What was the magnitude of the earthquake recorded in South America?

Solution: We form an equation with the data given in the first sentence:

$latex M_{C}=\log \left( \frac{I_{C}}{S}\right)=8.3$

$latex 8.3=\log \left( \frac{I_{C}}{S}\right)$

Now, we use the data from the second sentence to form the second equation:

$latex M_{SA}=\log \left( \frac{I_{SA}}{S}\right)$

$latex M_{SA}=\log \left( \frac{4I_{C}}{S}\right)$

Now, we solve for $latex M_{SA}$:

$latex M_{SA}=\log \left( \frac{4I_{C}}{S}\right)$

$latex =\log (4I_{C})-\log (S)$

$latex =\log (4)+\log (I_{C})-\log (S)$

$latex =\log (4)+(\log (I_{C})-\log (S))$

$latex =\log (4)+\frac{\log (I_{C})}{\log (S)}$

$latex =\log (4)+8.3$

$latex =0.602+8.3$

$latex =8.902$

$latex M_{SA}=8.9$

Therefore, the intensity of the earthquake in South America was 8.9 on the Ritcher scale.

## Chemical buffer

Chemical systems known as buffer solutions or chemical buffers have the ability to adapt to small changes in acidity to maintain a range of pH values. Buffer solutions have a wide variety of applications from aquarium maintenance to regulating pH levels in the blood.

### EXAMPLE

Blood is a regulatory solution. When carbon dioxide is absorbed into bloodstreams, it produces carbonic acid and lowers pH levels. The body compensates by producing bicarbonate, which is a weak base, to neutralize the acid.

The equation Henderson-Hasselbalch can be used to calculate the pH of a buffer solution. Hasselbalch was studying the carbon dioxide that dissolves in the blood and the model of the pH of the blood in this situation is $latex \text{pH}=6.1+\log \left( \frac{800}{x} \right)$, where x is the partial pressure of carbon dioxide in the arteries, measured in torr.

Find the partial pressure of carbon dioxide in the arteries if the pH is 7.2.

Solution: We use $latex \text{pH}=7.2$ in the given logarithmic equation and we get:

$latex 7.2=6.1+\log \left( \frac{800}{x} \right)$

$latex 1.1=\log \left( \frac{800}{x} \right)$

By solving this for x, we find:

$latex x=\frac{800}{{{10}^{1.1}}}=63.55$

Therefore, the partial pressure of carbon dioxide in the arteries is 63.55 torr.

## Information entropy

Another application of logarithmic functions is the entropy of information. The entropy of information H, in bits, of a randomly generated password consisting of L characters is given by $latex L \log_{2}(N)$, where N is the number of possible symbols for each character in the password.

In general, the larger the entropy, the stronger the password.

### EXAMPLE

• If an 8-character password is case-sensitive, that is, upper and lower case letters are considered different characters, it is composed only of letters and numbers, find the entropy of the information.

Solution: There are 26 letters in the alphabet, 52 if uppercase and lowercase are counted separately. There are 10 digits from 0 to 9. This equals a total of $latex N=61$ symbols. Since the password must be 8 characters, we have $latex L=8$. Therefore:

$latex H=8\log_{2}(61)$

$latex H=\frac{8\ln(61)}{\ln(2)}=47.44$

• How many symbols per character do we need to produce a 6-character password with 40-bit entropy?

Solution: We have $latex L=6$ and $latex H=40$, and we have to find N. Therefore, we have:

$latex 40=6\log_{2}(N)$

⇒    $latex N={{2}^{\frac{40}{6}}}=101.6$

Therefore, we would need 102 characters to get a password with 40-bit entropy.