P07 Growing with time

P07Growing With Time

A113 Mathematics

Scenario

John was researching on the growth in quantity of a certain type of micro-organism.

Table 1

Time (hours)Quantity (Q)01110211031,092411,0365109,60061,145,000

He recorded his experimental findings on their growth in quantity over a period of time, in Table 1.

He plot out a graph of the quantity of the micro-organism against time, in Graph 1.

From Graph 1, he realised that it was difficult for him to make accurate estimations of the quantity of the micro-organism at certain time periods:

between 0 to 4 hours or

beyond his data collection time period (e.g. after 6 hours)

Scenario

Graph 1 :

Quantity of micro-organism against Time

01234561101101092110361096001145000

Time (hours)

Quantity

Time (hours)lg Q001122.04133.03844.04355.04066.059

Table 2

Scenario

He applied the logarithmic scale to the quantity of micro-organism, as shown in Table 2 and plotted Graph 2.

Task 1:

Identify the benefits of representing Johns data in Graph 2 as compared to that of Graph 1.

Graph 2 :

lg (Quantity) against Time

0123456012.04139268515822273.03822263836871684.04281169180714715.03981055414834966.0588054866759045

Time (hours)

lg (Quantity)

Task 2:

Given thatlg Q = 1.0092 t + 0.004,

Investigate the time at which the quantity of the micro-organism will reach 2000.

Task 3:

If a small amount of a certain substance was supplied to the micro-organisms at optimal temperature, the growth of the micro-organism increases and the quantity of the micro-organism can be described as

lg Q = 3 log210t + log2102t

Investigate the time at which the quantity of the micro-organism will reach 2000.

Scenario

Learning Objectives

Define a logarithmic function

Apply laws of logarithm to solve equations involving logarithmic and exponential functions:

Identify graphs of exponential and logarithmic functions

Recognize the benefits of representing an exponential data in a logarithmic scale

An exponential growth is one in which the proportion in growth for every fixed time interval is the same.

In Table 1 of Johns collected data, the quantity of micro-organism increases by a factor of 10 for every 1 hour.

From 2 to 3 hours, the quantity increased by a factor of 1,092/110 10

From 3 to 4 hours, the quantity increased by a factor of 11,036/1,092 10

In other words, the growth of the micro-organism resembles an exponential growth.

Time(hours)Quantity (Q)Number of times the quantity increased 01-110= 10/1 = 102110= 110/10 1031,092= 1,092/110 = 9.927 10411,036= 11,036/1,092 =10.106 105109,600= 109,600/11,036 = 9.931 1061,145,000= 1,145,000/109,600=10.447 10

Exponential Growth

After applying logarithmic scale to the quantity of micro-organism in Table 1, the values in Table 2 are obtained.

Introduction to Logarithm

Table 1

Time (hours)Quantity (Q)01110211031,092411,0365109,60061,145,000Time (hours)lg Q001122.04133.03844.04355.04066.059

Table 2

Graph 1 :

Quantity of micro-organism against Time

01234561101101092110361096001145000

Time (hours)

Quantity

Graph 2 :

lg (Quantity) against Time

0123456012.04139268515822273.03822263836871684.04281169180714715.03981055414834966.0588054866759045

Time (hours)

lg (Quantity)

What is Logarithm?

Introduction to Logarithm

In the following logarithmic function:

x is the __________ of y to the ______ b

(where b > 0, b 1, y > 0)

This means that x is the ________ to which b must be raised to produce y:

Definition of Logarithmic Function

Commonly used values for log base are 10 and e, where e 2.71828 (mathematical constant)

For example, to evaluate or ,

We press in the calculator to get 0.30

Definition of Logarithmic Function

Logarithmic expressionStandard notationCalculatorCommon LogarithmNatural Logarithm

log

ln

log

2

Using calculator, evaluate the following logarithmic expressions:

Test Yourself

Changing Logarithmic form to Exponential form

Logarithm of Base and Logarithm of 1

Power Law

Product Law and Quotient Law

Change of Base

Laws of Logarithm

Changing Logarithmic form to Exponential form

The logarithmic form can be changed to the exponential form as follow:

Example:

Given that , find the value of x.

Given that , find the value of x.

Test Yourself

Given that , find the value of y.

Since ,

will result in:

will result in:

These are important results in logarithm functions.

Logarithm of 1 & Logarithm of Base

Evaluate the following logarithmic expressions:

Test Yourself

Note:

Product Law:

Quotient Law:

Example:

Evaluate

Product Law & Quotient Law

Given that , find the value of A.

Test Yourself

Given that , find the value of B.

Example:

Given that , find the value of p.

Power Law

x times

Given that , find the value of x.

Test Yourself

Given that , find the value of y.

To change the base of a logarithmic function from b to new base m:

This allows us to use a calculator to evaluate a log to any base other than 10 or e.

Example:

Change of Base

Evaluate

Test Yourself

Using Logarithm to Solve Equation

Consider the following equation:

To solve for x, you have to find out what exponent 3 has to be raised to produce 5

Since and , x is not an integer

Hence, logarithm needs to be used to find x:

Take common logarithm of both sides

Apply Power Law

Using Logarithm to Solve Equation

Consider another equation involving e:

Recall that _________

Hence, taking _______________ of both sides results in a simpler equation to solve for x:

Given that , find the value of x.

Test Yourself

Given that , find the value of x.

Exponential and Logarithmic Graphs

For an exponential function (where a > 1, a 1):

y is always positive

When x = 0,

For a logarithmic function (where a > 1, a 1):

x is always positive

When x = 1,

y

x

(0, 1)

y

x

(1, 0)

Learning Points

Changing Logarithmic form to Exponential form Logarithm of 1Logarithm of BaseProduct LawQuotient LawPower LawChange of Base

Define a logarithmic function

Laws of logarithm

Identify graphs of exponential and logarithmic functions

One-Minute Write

Please produce a written response of either of the following in only one minute:

Identify what you found the most complex point is from the seminar, or

Write down a question you have with respect to the concepts learnt so far.

quantity

previous

quantity

current

=

y

x

b

log

=

x

b

y

=