Logarithms A Quick Review of Exponents Exponent 5 7 Base - PowerPoint PPT Presentation

Logarithms

A Quick Review of Exponents Exponent 5 7 Base Exponents have two parts: 1. Base: The number of variable being multiplied 2. Exponent (or power or index): number of times it is multiplied 5 7 = 5 · 5 · 5 · 5 · 5 · 5 · 5 = 78125 a 3 = a · a · a

Introduction Now let’s consider a simple exponent: 2 3 = 8 This exponent says that 2 (the base) raised to the power of 3 (the exponent) gives us 8. So, if I were to ask the question: � What power of 2 gives 8? We know the answer is 3.

Intro, continued Here’s another question: � What power of 3 gives 81? We could write this question as 3 ? = 81 or, we could use “power” notation: power 3 ( 81 ) = ? ( What power of base 3 yields 81? ) The answer is 4: 3 4 = 81 or power 3 ( 81 ) = 4 power 3  3 4  = 4 or

Logarithm Definition Here are more examples power 2 ( 16 ) = 4 power 5 ( 125 ) = 3 power 1 / 2 ( 4 ) = - 2 Mathematicians do not use the word “power” in this context, they prefer the word “logarithm” and abbreviate it as “log” log 2 ( 16 ) = 4 log 5 ( 125 ) = 3 log 1 / 2 ( 4 ) = - 2

A Logarithm Pattern log 2 ( 16 ) = 4 = log 2  2 4  log 5 ( 125 ) = 3 = log 5  5 3  log 1 / 2 ( 4 ) = - 2 = log 1 / 2 ( 1 / 2 ) - 2  Logarithms undo exponents! log b ( b a ) = a log 10  10 5  = 5 Example : (Challenge) This also works in reverse: 10 log 10 ( 100 ) = 10 log 10  10 2  = 10 2 = 100 b log b ( a ) = a Example :

Why Logarithms Matter: Exponential Growth and Decay Example: growth of bacteria in a petri dish B = 25 · 10 h where B is the number of bacteria after h hours

Exponential Growth and Decay One of the questions we could ask about this growth is: When does the number of bacteria equal 100,000? Graph says: ≈ 3.6 hours How can we get an exact answer? 100000 = 25 · 10 h 10 h = 4000 Use logarithms (base 10) log 10  10 h  = log 10 ( 4000 ) h = log 10 ( 4000 ) Using a calculator: h = 3.602059991 hours

Why Logarithms Matter: Sound Volume The human threshold of hearing (ToH) is 10 - 12 W / m 2 . The threshold of hearing pain is 10 W / m 2 . Range is very large so a logarithmic scale is used � ToH is assigned 0 (zero) Bels � A sound that is 10 times more intense is 1 Bel (or 10 decibels) � A sound that is 100 times more intense is 2 Bels (20 decibels)

Sound Volume Intensity Intensity Level No. of Times W / m 2 Source ( dB ) Greater than ToH 1. × 10 - 12 10 0 Threshold of Hearing ( TOH ) 0 1. × 10 - 11 10 1 Rustling Leaves 10 1. × 10 - 10 10 2 Whisper 20 1. × 10 - 6 10 6 Normal Conversation 60 1. × 10 - 5 10 7 Busy Street Traffic 70 1. × 10 - 4 10 8 Vacuum Cleaner 80 6.3 × 10 - 3 10 9.8 Large Orchestra 98 1. × 10 - 2 10 10 iPod at Maximum Level 100 1. × 10 - 1 10 11 Front Rows of Rock Concert 110 1. × 10 1 10 13 Threshold of Pain 130 1. × 10 2 10 14 Military Jet Takeoff 140 1. × 10 4 10 16 Instant Perforation of Eardrum 160

Sound Volume Example A mosquito’s buzz is often rated with a decibel rating of 40 dB. Normal conversation is often rated at 60 dB. How many times more intense is normal conversation compared to a mosquito’s buzz? A mosquito’s buzz is 10 4 times greater than the ToH and the normal conversation is 10 6 times greater, so the ratio is: 10 6 10 4 = 10 2 = 100 Therefore, a normal conversation is 100 times more intense than a mosquito’s buzz. Another way of getting to this answer is to note that the difference in intensity is 20 dB or 2 Bels: 10 2 = 100

Why Logarithms Matter: Acids, Bases and the pH Scale Water Chemistry: Hydroxide ions, OH - and hydrogen ions, H + : � Acids donate hydrogen ions � Bases donate hydroxide ions Fact: a strongly acidic solution can have 100,000,000,000,000 times more hydrogen ions than a strongly alkaline solution. log 10 ( 100000000000000 ) = log 10  10 14  = 14 The pH scale goes from 0 (most acidic solution) to 14 (most alkaline solution)

Acids, Bases and the pH Scale H + Concentration pH Value Relative to Pure Water Example 0 10000000 battery acid 1 1000000 gastric acid 2 100000 lemon juice,vinegar 3 10000 orange juice,soda 4 1000 tomato juice,acid rain 5 100 black coffee,bananas 6 10 milk,saliva 7 1 pure water 8 0.1 sea water,eggs 9 0.01 baking soda 10 0.001 Great Salt Lake, milk of magnesia 11 0.000 1 ammonia solution 12 0.000 01 soapy water 13 0.000 001 bleach,oven cleaner 14 0.0000 001 liquid drain cleaner

Why Logarithms Matter: Earthquakes The Richter Scale, used to measure the energy of an earthquake, is a logarithmic scale (base 10). � Magnitude of 5 is 10 times greater than magnitude of 4 � Largest earthquake: Chile, 1960: 9.5 � Recent California quake: 7.1 10 9.5 10 7.1 = 10 9.5 - 7.1 = 10 2.4 = 251.189

Earthquakes If the Chilean earthquake had a magnitude of 9.5, what magnitude of earthquake is 500 times less? The question is asking us to find m : 10 9.5 10 m = 500 If we rearrange and use logs, we can get the answer: 10 m = 10 9.5 500 log ( 10 m ) = log 10 9.5 500 m = 6.801

Why Logarithms Matter: Large-Magnitude Math Try 500! on a calculator. 500 ! = 500 · 499 · 498 · 497 · ⋯ · 3 · 2 · 1 Interesting log property (we’ll prove it soon): log 10 ( 500 !) = log 10 500 + log 10 499 + log 10 498 + log 10 497 + ⋯ + log 10 3 + log 10 2 + log 10 1 Can easily use a spreadsheet to get: log 10 ( 500 !) = 1134.086409 If we undo the logarithm with an exponent we get: 500 ! = 10 1134.086409 = 10 1134 + 0.086409 = 10 1134 · 10 0.086409 = 1.220136826 × 10 1134

Common Logarithms Logarithms to the base 10 are called common logarithms (presumably because our counting system is base 10 and it’s pretty common). So when we write a logarithm statement and there is no base specified, we automatically assume it is in base 10): log ( x ) = log 10 ( x ) Most calculators will have a “log” or “Log” or “LOG” button. This button will give logs in base 10.

Natural Logarithms Exponents with a base of e occur frequently in the natural world. This means we often encounter functions of the form: f ( x ) = a e x f ( x ) = a e - x or where e = 2.71828182846... For example: log e ( 25 ) = 3.218875825 Since natural logarithms occur frequently in mathematical analysis, the symbol log e is given its own separate notation: ln This is pronounced as “ellen” in the US and as “lawn” in Canada.

Graphs of Logarithmic Functions

A Special Case: log b ( 1 ) All of the plots in the above graph intersect at ( 1, 0 ) . Why? To answer that question, let’s go back to our original “power” notation for logs: power b ( 1 ) = ? In words: what power of b gives us 1? b ? = 1 The answer, of course, is 0: b 0 = 1 We can write this as: log b ( 1 ) = 0

Domain and Range of log b ( x ) From the graphs, it should be clear that the domain of log b ( x ) is D = { x : x > 0 } = ( 0, ∞) The domain of the log b ( x ) function must be restricted to be greater than zero: there is no real-numbered exponent that will yield a negative number: b ? = - 1 In other words, power b ( x ) = log b ( x ) = undefined, if x < 0 From the graph, the range is R = { y : -∞ < y < ∞} = (-∞ , ∞)

The Product Property One property of exponents is how the exponents add: b p b q = b p + q If we take the logarithm of both sides of that equation we get: log b ( b p b q ) = log b ( b p + q ) The right-hand side of this equation reduces to: log b ( b p b q ) ⩵ p + q Let’s let m = b p and n = b q , so we get: log b ( m · n ) = p + q But, if we take the log of m and the log of n we get log b ( m ) = log b ( b p ) = p log b ( n ) = log b ( b q ) = q and This means that log b ( m · n ) = log b ( m ) + log b ( n )

The Quotient Property Another property of exponents is: b p b q = b p - q If we repeat the process that we used for the Addition Property we can write: b p = log b ( b p - q ) = p - q log b b q Using the same definitions of m and n we get m log b = p - q n So m log b = log b ( m ) - log b ( n ) n

The Power Property Another property of exponents is: ( b p ) q = b p · q Again, let’s take the logs of both sides of this equation: log b (( b p ) q ) = log b ( b p · q ) = p · q Let’s also use the same definitions of m : m = b p As before: log b ( m ) = p So log b ( m q ) = p · q = q · p And log b ( m q ) = q log b ( m )

Change of Base If you take a look at calculator, it (most likely) only has two log buttons: one for common logs (base 10) and one for natural logs (base e ). In some problems involving logs, the base is not 10 or e but you still need to calculate its value. For example, log 4 ( 17 ) = ? To calculate this, we want to change the base from 4 to either 10 or e . In other words, we want to write our log expression into the form: log 4 ( 17 ) = k · log ( 17 ) or j · ln ( 17 ) where j and k are “correction factors” to give us the right answer. Let’s do an example to see the pattern...

Change of Base Example Evaluate: y = log 4 ( 17 ) Let’s get rid of the base 4 by using exponents: 4 y = 4 log 4 ( 17 ) = 17 Now lets take the common log of both sides: log ( 4 y ) = log ( 17 ) If we use the Power Property we get y log ( 4 ) = log ( 17 ) If we divide both sides by log(4) we get our answer: y = log 4 ( 17 ) = log ( 17 ) = 2.0437 log ( 4 ) In this example, k = 1 / log ( 4 ) . Be sure to notice where the number 4 came from: it’s the original base.