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1 Higher Mathematics CfE Edition This document w as produced specially for the website, and we require that any copies or derivative works attribute the work to Higher Still Notes. For more details about the copyright on these notes, please see hsn . Exponentials and Logarithms Contents Exponentials and Logarithms 1 1 Exponentials EF 1 2 Logarithms EF 3 3 Laws of Logarithms EF 3 4 Exponentials and Logarithms to the Base e EF 6 5 exponential and Logarithmic Equations EF 7 6 Graphing with Logarithmic Axes EF 10 7 Graph Transformations EF 14 Higher Mathematics Exponentials and Logarithms Page 1 CfE Edition hsn . Exponentials and Logarithms 1 Exponentials EF We have already met exponential functions in the notes on Functions and A function of the form ( )xfx a= , where 0a> is a constant, is known as an exponential function to the base a. If 1a> then the graph looks like this: This is sometimes called a growth function.

2 If 01a<< then the graph looks like this: This is sometimes called a decay function. Remember that the graph of an exponential function ( )xfx a= always passes through ( ) 0, 1 and ( ) 1,a since: ( )001fa= =, ( )11faa= =. O 1 () 1,ayx, 01xyaa= <<O 1 ( ) 1,ayx, 1xyaa=> Higher Mathematics Exponentials and Logarithms Page 2 CfE Edition hsn . EXAMPLES 1. The otter population on an island increases by 16% per year. How many full years will it take the population to double? Let 0u be the initial population. ()()102210023320001 16 (116% as a decimal)1 1 61 1 6 1 1 61 1 61 1 61 1 6 1 1 61 1 6 1 1 uuuu uuu======== For the population to double after n years, we require 02nuu . We want to know the smallest n which gives 1 1 6n a value of 2 or more, since this will make nu at least twice as big as 0u. Try values of n until this is satisfied. 2345I f 2 , 1 1 61 3 52I f 3, 1 1 61 5 62I f 4 , 1 1 61 8 1 2I f 5 , 1 1 62 1 02nnnn== <== <== <== > On a calculator: 1 1 6 = 1 1 6 ANS = = Therefore after 5 years the population will double.

3 2. The efficiency of a machine decreases by 5% each year. When the efficiency drops below 75%, the machine needs to be serviced. After how many years will the machine need to be serviced? Let 0u be the initial efficiency. ()()102210023320000 95 (95% as a decimal)0 950 95 0 950 950 950 95 0 950 950 uuuu uuu======== When the efficiency drops below 00 75u (75% of the initial value) the machine must be serviced. So the machine needs serviced after n years if 0 950 75n . Higher Mathematics Exponentials and Logarithms Page 3 CfE Edition hsn . Try values of n until this is satisfied: 23456If 2, 0 950 903 0 75If 3, 0 950 857 0 75If 4, 0 950 815 0 75If 5, 0 950 774 0 75If 6, 0 950 735 0 75nnnnn==>==>==>==>==< Therefore after 6 years, the machine will have to be serviced. 2 Logarithms EF Having previously defined what a logarithm is (see the notes on Functions and Graphs) we now look in more detail at the properties of these functions.

4 The relationship between logarithms and exponentials is expressed as: log yayxxa= = where , 0ax>. Here, y is the power of a which gives x. EXAMPLES 1. Write 35125= in logarithmic form. 355125 3 log 125= =. 2. Evaluate 4log 16. The power of 4 which gives 16 is 2, so 4log 16 2=. 3 Laws of Logarithms EF There are three laws of logarithms which you must know. Rule 1 ( )logloglogaaaxyxy+= where ,, 0ax y>. If two logarithmic terms with the same base number (a above) are being added together, then the terms can be combined by multiplying the arguments (x and y above). EXAMPLE 1. Simplify 55log 2 log 4+. ()5555log 2 log 4log 2 4log 8.+= = Higher Mathematics Exponentials and Logarithms Page 4 CfE Edition hsn . Rule 2 ( )logloglogaa axyxy = where ,, 0ax y>. If a logarithmic term is being subtracted from another logarithmic term with the same base number (a above), then the terms can be combined by dividing the arguments (x and y in this case).

5 Note that the argument which is being taken away (y above) appears on the bottom of the fraction when the two terms are combined. EXAMPLE 2. Evaluate 44log 6 log 3 . ( )1244446312log 6 log 3loglog 2 (since 442). ==== = Rule 3 loglognaaxn x= where ,0ax>. The power of the argument (n above) can come to the front of the term as a multiplier, and vice-versa. EXAMPLE 3. Express 72 log 3 in the form 7loga. 72772 log 3log 3log Squash, Split and Fly You may find the following names are a simpler way to remember the laws of logarithms. ( )logloglogaaaxyxy+= the arguments are squashed together by multiplying. ( )logloglogaa axyxy = the arguments are split into a fraction. loglognaaxn x= the power of an argument can fly to the front of the log term and vice-versa. Higher Mathematics Exponentials and Logarithms Page 5 CfE Edition hsn . Note When working with logarithms, you should remember: log 1 0a= since 01a=, log1aa= since 1aa=.

6 EXAMPLE 4. Evaluate 73log 7 log 3+. 73log 7 log 3112.+= += Combining several log terms When adding and subtracting several log terms in the form logab, there is a simple way to combine all the terms in one step. Multiply the arguments of the positive log terms in the numerator. Multiply the arguments of the negative log terms in the denominator. EXAMPLES 5. Evaluate 121212log 10 log 6 log 5+ . 1212121212log 10 log 6 log 510 6log5log 121.+ = == 6. Evaluate 66log 4 2 log 3+. ()6626666662log 4 2 log 3log 4 log 3log 4 log 9log 4 9log 362(since 636).+=+=+= === OR ()()()662666666666log 4 2 log 3log 22 log 32 log 2 2 log 32 log 2 log 32 log 2 32 log 62(since log 6 1).+=+=+=+= === ()12 log1212log log++10612log 5() logaarguments of positive log terms arguments of negative log terms Higher Mathematics Exponentials and Logarithms Page 6 CfE Edition hsn.

7 4 Exponentials and Logarithms to the Base e EF The constant e is an important number in Mathematics , and occurs frequently in models of real-life situations. Its value is roughly 2 718281828 (to 9 ), and is defined as: ()11 as nenn= + . If you try very large values of n on your calculator, you will get close to the value of e. Like , e is an irrational number. Throughout this section, we will use e in expressions of the form: xe, which is called an exponential to the base e; logex, which is called a logarithm to the base e. This is also known as the natural logarithm of x, and is often written as lnx ( lnlogexx). EXAMPLES 1. Calculate the value of log 8e..log 8 2 08 (to 2 ).e= 2. Solve log9ex=. 9log9so 8103 08 (to 2 ).exxex=== 3. Simplify ( )( )4 log 23 log 3eeee expressing your answer in the form loglogeeabc+ where a, b and c are whole numbers. ( )( )434 log 23 log 34 log 2 4 log3 log 3 3 log4 log 2 4 3 log 3 31 4 log 2 3 log 31 log 2log 31 log 16 log ee eeeeeeeeeeeee =+ =+ =+ =+ =+ OR ( )( )( )()( )()4343434 log 23 log 3log 2log 32log316log2716log27loglog 16 log 271 log 16 log eeeeeeeeeeeee = = = = =+ =+ On a calculator: ex 9 = On a calculator: ln 8 = Remember ( )=nnnabab.

8 Higher Mathematics Exponentials and Logarithms Page 7 CfE Edition hsn . 5 exponential and Logarithmic Equations EF Many mathematical models of real-life situations use exponentials and logarithms. It is important to become familiar with using the laws of logarithms to help solve equations. EXAMPLES 1. Solve log 13 loglog 273a aax+= for 0x>. log 13 loglog 273log 13log 27313273(since loglog) aaaaaaxxxxy xyx+==== == 2. Solve ()()1111log43log231xx+ = for 32x>. ()()()1111111log43log23143log123431111(s ince log)2343 11 2343 y xaxxxxxxx+ =+ = += == = += += == 3. Solve ()()()log 21log 310log 11aa app p++ = for 4p>. ()()( )()()( )()()()()22log 21log 310log 11log21 310log 1121 31011620310 11062810 03 1 50aa aaapp pppppppp ppppppp++ =+ = + = + = =+ = 133 10pp+== or == Since we require 4p>, 5p= is the solution.

9 Higher Mathematics Exponentials and Logarithms Page 8 CfE Edition hsn . Dealing with Constants Sometimes it may help to write constants as logs to solve equations. EXAMPLE 4. Solve 22log 7log3x=+ for 0x>. Write 3 in logarithmic form: 223223 313 log 2 (since log 2 1)log 2log ==== Use this in the equation: 2222278log 7loglog 8log 7log +=== OR ( )222227log 7log3log 7 + == Converting from log to exponential form: Solving Equations with unknown Exponents If an unknown value ( x) is the power of a term ( xe or 10x), and its value is to be calculated, then we must take logs on both sides of the equation to allow it to be solved. The same solution will be reached using any base, but calculators can be used for evaluating logs to the base e and 10. EXAMPLES 5. Solve 7xe=. Taking loge of both sides: OR Taking 10log of both sides: loglog 7loglog 7 ( log1)log 71 946 (to 3 ).

10 Xeeee eeexeexx===== 101010101010loglog 7loglog 7log 7log1 946 (to 3 ).xexexex==== Higher Mathematics Exponentials and Logarithms Page 9 CfE Edition hsn . 6. Solve 31540x+=. ()31log 5log 4031 log 5 log 40log 4031log 531 2 292031 29200 431 (to 3 ).xeeeeeexxxxx+=+=+=+=== exponential Growth and Decay Recall from Section 1 that exponential functions are sometimes known as growth or decay functions. These often occur in models of real-life situations. For instance, radioactive decay can be modelled using an exponential function. An important measurement is the half-life of a radioactive substance, which is the time taken for the mass of the radioactive substance to halve. EXAMPLE 7. The mass G grams of a radioactive sample after time t years is given by the formula 3100tGe =. (a) What is the initial mass of radioactive substance in the sample?