C2 – Logarithms
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C2 LogarithmsDr J FrostStarterSketch a graph of y = 3x (Note: this was once used as an exam question)11 mark: Two of the three criteria.2 marks: All of the three criteria.
Correct shape in left quadrant.Correct shape in right quadranty-intercept of 1.?Functions and their inversesSome operators exist to provide the opposite of others.4x 3x 34FunctionInverse4x + 3x - 344x2x44x55x4127161024????Functions and their inversesSome operators exist to provide the opposite of others.FunctionInverse43xlog3 x481?How therefore would describe the effect of log3 x in words?log3 xIt finds the power that, when 3 is raised to it, gives you x.
i.e. If y = log3 x, then 3y = x
It is the opposite/inverse of exponentiation. We describe this as the logarithm of x base 3 or log of x base 3 or taking the log of x base 3.log2 8 = 3Computing logsRemember that logarithms find the missing power.?Bro Tip #1: Imagine what power would slot in the middle of the two.log2 8 = 33=Click to start bromanimationlog3 9 = 2?log10 100 = 2?log4 1 = 0?Bro Tip #2: loga 1 = 0 (a > 0)log3 3 = 1?Bro Tip #3: loga a = 1 (a>0)log2( ) = -112Computing logsRemember that logarithms find the missing power.?Bro Tip #4: When we take the log of any value between 0 and 1 (exclusive), we end up with a negative number. log2( ) = -318log3( ) = -4181log4 (-1) = ??__i__loge 4?Bro Tip #5: If you want a real result, you can only take logs of positive values. -4 -2 2 4 6 8 10 12 148
y = log2 xx0.250.51248y-2-10123??????Click to brosketchGet students to sketch axes and tables in their books.7Using logsLogs help us solve equations when the power is unknown.Find the x for which 10x = 500We can write this as x = log10 500
Broculator Tip:The log button on your calculator is implicitly base 10. So [log]  will give you log10 500?log9 81 = 2log2 8 = 3?23 = 892 = 81Bro Tip: In both cases, the 2 is the base.log3 81 = 4?34 = 81log3 55 = x?3x = 55More on rewriting Powers as LogarithmsExercisesC2 Chapter 3 Pg 42 Exercise 3B (All questions)Exercise 3C (Q1, 3, 5, 7)Laws of LogsThese are 3 laws of logs that you need to remember. loga xy = loga x + loga yloga = loga x - loga yloga (xk ) = k loga xProving these involves rewriting the logs as exponential expressions, then using laws of indices. xy
log3 (a2b)log2 5log230 log262log3 a + log3 blog4 ( )?3log4 (a) 4log4 b3loga (ab)a3b4???Put in the form k + loga (..)Write the following as a single logarithm.Laws of Logs2loga b2loga b + 3logacloga(b2c3)4loga(b)1 + loga b?loga(ab)12?Now the other way round! Write in the form loga x, loga y and loga z.?-loga xloga( )1x?Laws of LogsUsing logs in scienceThe well-known Moores Law states that the processing power of computers doubles every two years. Its been remarkably accurate so far!
If we were to plot the number of transistors against the year, the type of graph would be exponential.The graph would look rubbish if we chose a range of values on the y-axis to accommodate all the values, because except for the last few years, most of the points would look close to 0.?Using logs in scienceThe well-known Moores Law states that the processing power of computers doubles every two years. Its been remarkably accurate so far!
This graph gets around the problem by letting the y-values increase by a factor of 10 for each unit, rather than increasing by a constant amount each time. Technically this is not allowed! Using logs in scienceThe well-known Moores Law states that the processing power of computers doubles every two years. Its been remarkably accurate so far!
We could instead use a logarithmic scale. We can take the log base 10 of these values. Then well get 3, 4, 5, 6, ..., which is now allowed!*log10(Number of transistors)345678910Logarithmic scales turn exponential graphs into linear ones (i.e. a straight line), thus making it much easier to plot all the points together.* Although realistically, a scale of 10, 100, 1000, etc. is permissible as long as were mindful that its a logarithmic scale.
Using logs in scienceLogarithmic scales are used for earthquakes and noise levels.From our laws of logs, in base a... loga ax = 1 + loga xwhen a quantity gets a times bigger,the overall result only increases by 1.Thus using logarithms turns a factor difference into a constant difference.The Richter Scale is used to measure the magnitude of earthquakes. The scale is logarithmic (base 10): it means if amplitude of the earthquakes waves gets 10 times bigger, the value on the Richter Scale only increases by 1.
Earthquakes of magnitude 6 vs 7 doesnt look like a substantial difference, but just the one point difference means its ten times worse! ExercisesC2 Chapter 3 Pg 42 Exercise 3D (All questions)We saw how we can solve equations like 10x = 125.But what about when the base is different, e.g. 3x = 20?OPTION 1: The Look at me, I have a fancy calculator methodx = log3 20??OPTION 2: The change of base method3x = 20log10 3x = log10 20?Super Bro Tip:Whenever youre trying to solve an equation where the variable appears in the power, your first instinct should always be TAKE LOGS DAMMIT!x log10 3 = log10 20x =log10 20log10 3??Solving ax = bWe saw a second ago that we could change the base to find log3 in terms of log10.x = log3 203x = 20x =log10 20log10 3METHOD 1METHOD 2More generally, to change the base from a to b:loga x = logb xlogb a Click to start bromanimationxaChanging the Baselog10 5log10 2log2 5 in base 10?Express these logarithms in the specified new base.log12 10log12 7log7 10 in base 12log9 5log9 10log10 5 in base 9log10 10log10 5log5 10 in base 10___1___log10 5=???Broculator Tip:This is how you could find log25 on a calculator if you didnt have the fancy extra log button. i.e. Change to base 10!
All your base belong to usBro Tip: When you switch the argument and base, you take the reciprocal.loga b = ___1___logb aAll your base belong to usVariables appear in powers, so apply Bro Tip.(The base of the log doesnt matter)log 7x+1 = log 3x+2x =2log 3 log 7log 7 log 37x+1 = 3x+2(x+1)log 7 = (x+2)log 3xlog 7 + log 7 = xlog 3 + 2log 3xlog 7 - xlog 3 = 2log 3 log 7x(log 7 - log 3) = 2log 3 log 7Solving Equations involving Variables in PowersBro Tip:By recognising that 52x = (5x)2, weve turned the equation into a quadratic!(5x)2 + 7(5x) 30 = 052x + 7(5x) 30 = 0Let y = 5xy2 + 7y 30 = 0(y+10)(y-3) = 0y = -10 or y = 35x = -10 or 5x = 3x = log5(-10) or x = log53Cant have log of a negative number, so not a real solution.Solving Equations involving Variables in Powers22x + 3(2x) 4 = 03x-1 = 8x+1 x = -3.24x = 0??Solve log2 (2x + 1) log2 x = 2When solving, you can often either:Get in the form logab = c. Then rearrange as ac = bGet in the form logab = logac. Then b = c.EdExcel exam questions:Solve log 2 (x + 1) log 2 x = log 2 7Exam TechniqueAnswer: x = 1/2Answer: x = 1/6??26When solving, you can often either:Get in the form logab = c. Then rearrange as ac = bGet in the form logab = logac. Then b = c.Edexcel exam questions:log2 (11 6x) = 2 log2 (x 1) + 3Exam TechniqueAnswer: x = 8 or 1/8Answer: x = -1/4 or 3/2??