Question

Search the internet for real-world applications of logarithms or exponential functions. Share the application you discovered with the class along with how it relates to logarithms or exponential functions. Besure to cite the resource. What is the relationship between exponentials and logarithms?

Answer 1

Apploications involving logarithmic functions.

Much of the power of logarithms is their usefulness in solving exponential equations. Some examples of this include sound (decibel measures), earthquakes (Richter scale), the brightness of stars, and chemistry (pH balance, a measure of acidity and alkalinity).

1.Let’s look at the Richter scale, a logarithmic function that is used to measure the magnitude of earthquakes. The magnitude of an earthquake is related to how much energy is released by the quake. Instruments called seismographs detect movement in the earth; the smallest movement that can be detected shows on a seismograph as a wave with amplitude A0.
A – the measure of the amplitude of the earthquake wave
A0 – the amplitude of the smallest detectable wave (or standard wave)
From this you can find R, the Richter scale measure of the magnitude of the earthquake using the formula

2.Sound is measured in a logarithmic scale using a unit called a decibel. The formula looks similar to the Richter scale:

where P is the power or intensity of the sound and P0 is the weakest sound that the human ear can hear.

3.The measure of acidity of a liquid is called the pH of the liquid. This is based on the amount of hydrogen ions (H+) in the liquid. The formula for pH is:
  pH = −log[H+]
where [H+] is the concentration of hydrogen ions, given in a unit called mol/L (“moles per liter”; one mole is 6.022 x 1023 molecules or atoms).

Apploications involving Exponential functions.

Exponential functions are used for even more contexts, including population and bacterial growth, radioactive decay, compound interest, cooling of objects, and growth of phenomena such as virus infections, Internet usage, and popularity of fads.
1.For example, recall that the formula for compound interest is
,
where P is principal, A is amount, r is the annual rate, m is the number of compounding periods, and t is the number of years.

Relationship between Logarthimic and Exponential function

Logarithms are the "opposite" of exponentials, just as subtraction is the opposite of addition and division is the opposite of multiplication. Logs "undo" exponentials. Technically speaking, logs are the inverses of exponentials.

Exponential Function=> y=bx
=> log(y)=xlog(b)
=> logby=x
=> Logarthimic Function