Application of Logarithmic Functions to Science

MHF4UIMonday October 2nd , 2012

The General Formula for Problems of Exponential Growth and Decay

The standard form of an exponential growth or decay is:

Where, is the initial quantity at t = 0b is the factor of growth (1 + i) or decay (1 – i)i is the percent rate of growth or decay (must be expressed as a decimal)

t is the number of growth or decay periodsA(t) is the quantity at time t

Half-Life Example

Homework 7.2 Part A Question 8

A 20-mg sample of thorium-233 decays to 17mg after 5 minutes.a) Determine the half-life of thorium-233.

Half-Life (Solution)A 20-mg sample of thorium-233 decays to 17mg after 5 minutes.

a) Determine the half-life of thorium-233. = b = t = A(t) =

Half-Life (Solution)

A 20-mg sample of thorium-233 decays to 17mg after 5 minutes.a) Determine the half-life of thorium-233.

Now we can calculate the half life of thorium-233.(Remember that half-life is a length of time)

= b = (1-i) = (1-0.0320)t = A(t) =

Application of Logarithmic Functions to Physical Sciences

We can use Logarithmic functions in science to compare variables that occur over large orders of magnitude

Said another way, we want a universal way to compare very small values to very large values

Logarithmic Scales are often used in these types of situations. Some examples of Logarithmic Scales include:

• Decibel Scale• Richter Scale• pH Scale

Applications to of Logarithms to Sound

We are interested in measuring the intensity and loudness of soundSound is measured in Decibels (L)

The formula for measuring the intensity of sound is:

Where,L is the loudness of the sound (Decibels) is the intensity of the subject sound is the intensity of sound at the threshold of hearing

The Decibel Scale

Sound Decibels

Threshold of Hearing (TOH) 0 dB

Rustling Leaves 10 dB

Whisper 20 dB

Normal Conversation 60 dB

Busy Street Traffic 70 dB

Large Orchestra 98 dB

Front Rows of Rock Concert 110 dB

Threshold of Pain 130 dB

Military Jet Takeoff 140 dB

Instant Perforation of Eardrum 160 dB

Lung rupture 200 dB

Decibel Scale Example 1

A sound is 1,000,000 times more intense than a sound at the threshold of hearing. What is its loudness in decibels?

L = =

Decibel Scale Example 2

How many more times intense is the sound heard from the Front Rows Seats at a Rock Concert (110dB) vs. normal conversation (60dB)?

Applications to of Logarithms to Natural Disasters

We are interested in measuring the intensity and magnitudes of earthquakes

The Richter Scale is used to measure Earthquakes

The formula for comparing magnitudes of Earthquakes is:

Where,M is the magnitude of the Earthquake is the intensity of the subject Earthquake is the intensity of a reference (comparison) Earthquake

March 11th, 2011

Thousands of homes were destroyed, many roads were impassable, trains and buses were not running, and power and cellphones remained down.

Magnitude Scale Example

The earthquake in Japan measured 8.9 on the Richter Scale. Compare this earthquake to the one that occurred in Leamington,

Ontario that measured 3.0 on the Richter Scale.

Applications to of Logarithms to Chemistry

We are interested in measuring acidity or alkalinity of liquids.

The pH Scale is used to define if a solution an acid, base or neutral.

The formula for measuring pH is:

Where,is the concentration of hydrogen ions in a liquid in moles/litre.

Acid pH < 7

Neutral pH = 7

Base pH > 7

pH Scale Example 1

Coca-Cola has a pH 2.5, what is its hydrogen ion concentration?

pH =

pH Scale Example 2

Tomato Juice has a hydrogen ion concentration of about 0.0001.What is its pH? Is it an acid or a base?

pH =

Homework Questions:

• Textbook Chapter 6.5• Part A: 1,2,6,7,8• Part B: 9,10,12