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Thursday, September 11, 2014

Global warming and green house effect

 The heating up of the earth surface due to trapping of infrared radiations reflected by earth's surface by CO2 layer in the atmosphere is called green house effect.

Some major green house gases and their sources:


GasSources
Carbon Dioxide
(CO2)
1)Fossil fuel combination (coal,oil,natural gas)
2) Deforestation and land use changes
Methane
(CH4)
1) Enteric fermentation in cattle and insects.
2) Biomass burning and garbage land fills
3) Leakage of gases from coal mines and natural gas.
4) Rice paddies
5) Swamps and tundra
Chloro fluoro carbon
(CFC'S)
1) Plastic foams
2) Refrigeration and air-conditioning
3) Aerosols (spray propellants) CFC'
4) Industrial solvents
Nitrous oxide
(N2O)
1) Fertilizer use
2) Fossil fuel combustion
3) Biomass burning
4) Changing land use

Global warming and green house effect

An increase in average temperature  of the earth's atmosphere due to green house effect which can have effects on climate and key life, support system of planet.
An increase in global temperature is also likely to increase the incidences of infections diseases like malaria, dengue, sleeping sickness and yellow fever.

Computer model predicts that during next 50 years the average temperature rise from 1.5 0 C to 3.5 0C.
Such a change ,which is very rapid by natural standards ,could have far reaching effects , especially at extreme levels .These could be in the form of
1) a rise in sea levels up-to 5 to 6 meters from the melting of ice cap and glaciers and the thermal expansions of sea water.
2) altered climate pattern altering ecosystems and
3) more extremes of weather .

Wednesday, September 10, 2014

Logarithm


LogarithmDefinition Logarithm :
If m= a x, where a>0 ,a not equal to 1 , m>0 then x is called the logarithm of m to the base a and is written as x = log a m, where m, a, x are real numbers.

Laws of logarithms:
1. Law of Product: log (A.B) =log A +log B
2. Law of Quotient : log(A/B) =log A - log B
3. Law of Exponent : log m n = n log m

Properties:
1)We have m = a x if and only if x = log a m.
2)Logarithms of negative numbers and zero are not defined.
3) a0 =1 Hence, 0 = log a 1
i.e. logarithm of 1 to any base is always zero.
4) a1 =a Hence , 1 = log a a
i.e. logarithm of any number to the same base is 1.
5) x = log a ax
6) If log a m = log a n then m = n
7)If a>1 and m>n then log a m > log a n and conversely.

Write logarithm form:
1) 5 3 = 125
Solution:
3 = log 5 125   (By definition of logarithm)

2) 8 3 =512
Solution:
3 = log 8 512   (By definition of logarithm)


Express in exponential form:
1) log 2 128 = 7
Solution: 128 = 2 (By definition of logarithm)

2) log 9 6561 =4
Solution: 6561 = 2 (By definition of logarithm)


Simplify : 
1) log 5 + 2 log 4
Solution:
=log 5 + log 42 (By Law of Exponent)
=log 5 + log 16
=log (5 . 16) (By law of Product)
=log 80

2) 2 log 7 - log 14
Solution:
=log 7 2 - log 14 (By Law of Exponent)
=log 49 - log 14
=log (49/14) (By law of Quotient)
=log (7/2)

3)log 3 + log 2 - 2 log 5
Solution:
=log 3 + log 2 - log 52 (By Law of Exponent)
=log 3 + log 2 - log 25
=log (3 . 2) - log 25 (By Law of Product)
=log 6 - log 25
=log (6/25) (By Law of Quotient)

4) 2 log 3 - 1/2 log 16 + log 12
Solution:
=log 3 2 - log 16 1/2 + log 12 (By Law of Exponent)
=log 9 - log4 + log 12
=log (9/4) + log 12 (By Law of Quotient)
=log (9/4 .12) (By Law of Product)
=log (9. 3)
=log 27


Prove that:
1) log 540 = 2 log 2 + 3 log 3 + log 5
Solution:
L.H.S = log (540)
=log (4 X 27 X 5)
=log ( 22 X 33 X 5)
=log 22 + log 33 + log 5 (By Law of Product)
=2 log 2 + 3 log 3 +log 5 (By Law of Exponent)
=R.H.S.
Hence, log 540 = 2 log 2 + 3 log 3 + log 5

OR

R.H.S =2 log 2 + 3 log 3 + log5
=log 2 2 + log 33 + log 5 (By Law of Exponent)
=log 4 + log 27 + log 5
=log (4 X 27 X 5) (By Law of Product)
=log 540
=L.H.S.
Hence, log 540 = 2 log 2 + 3 log 3 + log 5

2) log 360 = 3 log 2 + 2 log 3 + log 5
R.H.S =3 log 2 + 2 log 3 + log5
=log 2 3 + log 32 + log 5 (By Law of Exponent)
=log 8 + log 9 + log 5
=log (8 X 9 X 5) (By Law of Product)
=log 360
=L.H.S.
Hence, log 360 = 3 log 2 + 2 log 3 + log 5

OR

L.H.S = log (360)
=log (8 X 9 X 5)
=log ( 23 X 32 X 5)
=log 23 + log 32 + log 5 (By Law of Product)
=3 log 2 + 2 log 3 + log 5 (By Law of Exponent)
=R.H.S.
Hence, log 360 = 3 log 2 + 2 log 3 + log 5


Evaluate:
1) log 81/log 9
=log 92 / log 9
=2 log 9 / log 9   (By Law of Exponent)
=2

2)log 2 5 / log2 11 - log4 5 / log4 11
Solution :
We know ,Change of Base formula is log a m = log m /log a

=log 5/log 2/log 11/log 2 - log 5/log 4/ log 11 /log 4
=log 5 . log 2 / log 11 . log 2 - log 5 .log 4 / log 11 .log 4
=log 5 / log 11 - log5 / log 11
=0

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