# 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

## 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.  