Exponential equations and logarithms

Exponential equations and logarithmsFun with exponential equationsEvery pair share a packet of M&Ms.You are supposed to firstly empty the M&Ms on the serviettes provided randomly. Count the number of M&Ms in all and record your reading as the zeroth reading. Then eat all the M&Ms facing up (i.e. with the letters on top) and count those remaining as the first reading.Now put the M&Ms back into the bag, shake and pour out again. Eat those facing up and count the remaining number of M&Ms again and record your reading as the second reading.Continue the procedure until you have no more M&Ms remaining. Fun with exponential equationsNow plot these points on Microsoft Excel, following my instructions. What kind of graph does your readings give? The functions that gives this kind of graphs are called exponential functions.Exponential equationsExponential GrowthExponential DecayThe Beggar and the King

http://raju.varghese.org/articles/powers2.html5The Beggar and the King Do you think the beggar made a wise request?

Was the King right in agreeing to the beggars request?

What do you think will happen?

The Beggar and the KingDayWeight of rice/ g1222x2=22=432x2x2=23=842x2x2x2=24=1652x2x2x2x2=25=3262x2x2x2x2x2=26=6472x2x2x2x2x2x2=27=1287Exponential Growth

The Beggar and the KingMoral of the story:Know your Mathematics! Use them in everyday life! Folding a piece of paperhttp://raju.varghese.org/articles/powers2.html

Exponential Decay of radioactive substancesCesium-137 and strontium-90 present long-term environmental hazards and can be absorbed throughout the body, particularly bones. Plutonium-239 exposure often leads to lung cancer, and it has a half-life of 24,000 years, so it would be around for a long, long time.(A half-life is the amount of time it takes for half of the radioactive isotopes in a substance to decay.)

Exponential decay

Graphs of exponential functions

y = 2xy = 3x

Graphs of exponential functions

Graphs of exponential functions

Looks the same? Why?Logarithm

IndexBase

Index formLogarithmic formFor any positive number a, except 1,

What you want to know is: a to the power of WHAT gives y?This WHAT is your x, which is what you want to find in the logarithmic form.LogarithmWhy cant a =1 or a < 0?a is the base of the logarithm. It cannot be equal to 1 as 1x is always 1. a cannot be negative as powers of negative numbers change sign.For logay to be defined:y > 0, why?a > 0, a 1

For any positive number a, except 1,

(-2)^2=?(-2)^2.1 =?17LogarithmFor logay to be defined:y > 0a > 0, a 1

The following are not defined:log12log-34log2(-1) LogarithmConvert to logarithmic form.Convert to index form.

What is the base here?What is the index here?

What is the index here?What is the base here?Convert the following to logarithmic form

Convert the following to index form

Solve the following equations

2 important conclusion

What conditions of a do you need to impose here?A>0 and a not equal to 122Evaluate the following

The Common LogarithmLogarithms with a base of 10 are called common logarithmsUse the LOG key on your calculator to find the values of common logarithms.log10x is often abbreviated as lgx.Uses: In chemistry as a measure of acidity, in earthquakes as a measure of the strength (Richter Scale) etc. For earthquakes, the wave amplitude is typically very big, so a common logarithm scale is used. Think: a 9.0 earthquake (Japan) is how many times more powerful (in terms of wave amplitude) than an 8.0 earthquake (Sichuan earthquake)? Solve the following equations, using your calculators, giving your answers correct to 4 s.f.

x = 6.459x = 1.364x = 1.883 or x=0.7384Skip firstNatural LogarithmThere is another logarithm to the base of a special irrational number called e, named after Leonhard Euler.e has a value of 2.7183.Logarithm to the base e, logex, is often abbreviated as ln x. ln x is called Natural logarithm, or Naperian logarithm (after John Napier)Can you find e and ln on your calculator?Where is it used? Radioactive decay, first order reaction in Chemistry, calculus etc.Solve the following equations

Skip firstLaws of logarithm

Note: m and n are positive and a > 0, a 1Product LawQuotient LawPower LawProofsProduct Law:

ProofsQuotient Law:

ProofsPower Law:

Are the following true?

Simplify the following

Logarithmic equations of the same baseFor two logarithms of the same base,

Solve the following equation

Solve the following equation

Solve the following simultaneous equations

Change of basesIf a, b and c are positive numbers and a1,c1, then:

Proof

ExampleEvaluate log75 log59 log97Hint: Change all the bases to common log!

ExampleEvaluate

Hint: Change all the bases to common log!

Ans: 8Solve the following equation

Examples (using all you have learnt)Given that log43 = a and log45 =b, express log445 in terms of a and b.

ExerciseExercise 2.4Qns 4e-h, 5d-f, 6, 7, 8, 9, 10Exercise 2.5Qns 2e-f, 3e-f, 4e-fExercise 2.6Qns 4c, 5d-f, 6a-d, 7, 8, 9, 10d-f, 11Exercise 2.7Qns 1e-h, 2, 3, 4a, 5c-e, 6c-d, 7, 8Equations of the form ax=bRemember these?

Can you solve the equations below like what you did?

Equations of the form ax=bIf you cannot express both sides of the equation with the same base, the strategy is to take LOGARITHM on both sides.Solve the following equations:

Example

x = 2.22

Graphs of y=e-x and y=ex

Graphs of lgx and lnx

Exercise 2.8Qns 1h-i, 3, 4b-c, 5b, 5d, 6b, 6c, 7a-b, 8