Logarithms - changing the base
This leaflet gives this formula and shows how to use it. A formula for change of base. Suppose we want to calculate a logarithm to base 2. The formula states.
mc logs
MATHEMATICS 0110A CHANGE OF BASE Suppose that we have
So we get the following rule: Change of Base Formula: logb a = logc a logc b. Example 1. Express log3 10 using natural logarithms. log3 10 =.
Change of Base
Appendix N: Derivation of the Logarithm Change of Base Formula
We take loga of each side of this equation which gives us loga by = loga x
Change of Base Formula.pdf
The Change of Base Formula. Use a calculator to approximate each to the nearest thousandth. 1) log3. 3.3. 2) log2. 30. 3) log4. 5. 4) log2. 2.1. 5) log 3.55.
Change of Base Formula
Logarithms.pdf
16 nov. 2017 The zero exponent rules can also be used to simplify exponents. ... This law allows a logarithm with a given base to be changed to a new ...
Logarithms
Logarithms – University of Plymouth
16 janv. 2001 Change of Bases ... following important rules apply to logarithms. ... (b) We can do the same calculation using instead logs to base e.
PlymouthUniversity MathsandStats logarithms
6.2 Properties of Logarithms
We apply the Change of Base formula with a = 3 and b = 10 to obtain 32 = 102 log(3). Typing the latter in the calculator produces an answer of 9 as required. 2.
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Sometimes it is useful to change the base of a logarithmic expression. For example the log key on a calculator is for common
NHTI Learning Center Math Lab G-25 Rules for Logs
4 janv. 2006 A log (base 10) of a number is a power you would put on 10 to ... Before we had scientific calculators logs were used to do calculations.
lcg rulesforlogs
Staff Regulations Rules and Instructions Applicable to Officials of
20 janv. 1992 of the country of which he/she is a national is taking up duty without changing country the provisions of Rule.
Staff Rules EN
Levelling-Up Basic MathematicsLogarithmsRobin HoranThe aim of this document is to provide a short, self assessment programme for students who wish to acquire a basic competence in the use of logarithms.Copyrightc 2000rhoran@plymouth.ac.ukLast Revision Date: January 16, 2001 Version 1.00
Table of Contents1.Logarithms2.Rules of Logarithms3.Logarithm of a Product4.Logarithm of a Quotient5.Logarithm of a Power6.Use of the Rules of Logarithms7.Quiz on Logarithms8.Change of BasesSolutions to QuizzesSolutions to Problems
Section 1: Logarithms 31. Logarithms (Introduction)LetaandNbe positive real numbers and letN=an:Thennis
called thelogarithm ofNto the basea. We write this asn= logaN:Examples 1(a)Since 16 = 24;then 4 = log216:(b)Since 81 = 34;then 4 = log381:(c)Since 3 =p9 = 912;then 1=2 = log93:(d)Since 31= 1=3;then1 = log3(1=3):
Section 1: Logarithms 4ExerciseUse the denition of logarithm given on the previous page to deter-mine the value ofxin each of the following.1.x= log3272.x= log51253.x= log2(1=4)4.2 = logx(16)5.3 = log2x
Section 2: Rules of Logarithms 52. Rules of LogarithmsLeta;M;Nbe positive real numbers andkbe any number. Then the
following important rules apply to logarithms.1:logaMN= logaM+ logaN2:logaMN= logaMlogaN
3:logamk=klogaM
4:logaa= 1
5:loga1 = 0
Section 3: Logarithm of a Product 63. Logarithm of a Product1. Proof thatlogaMN= logaM+ logaN:Examples 2(a)log64 + log69 = log6(49) = log636:
Ifx= log636;then 6x= 36 = 62:
Thus log
64 + log69 = 2:(b)log520 + log414= log52014:
Now 2014= 5 so log520 + log414= log55 = 1:Quiz.To which of the following numbers does the expression log315 + log306 simplify?(a)4(b)3(c)2(d)1
Section 4: Logarithm of a Quotient 74. Logarithm of a Quotient1. Proof thatlogaMN= logaMlogaN:Examples 3(a)log240log25 = log2405= log28:
Ifx= log28 then 2x= 8 = 23;sox= 3:(b)If log35 = 1:465 then we can nd log306:Since 3=5 = 06;then log306 = log335= log33log35:
Now log
33 = 1;so that log306 = 11465 =0465Quiz.To which of the following numbers does
the expression log212log234simplify?(a)0(b)1(c)2(d)4
Section 5: Logarithm of a Power 85. Logarithm of a Power1. Proof thatlogamk=klogaMExamples 4(a)Find log10(1=10000):We have 10000 = 104;so 1=10000 =
1=104= 104:
Thus log
10(1=10000) = log10104=4log1010 =4;where
we have used rule 4 to write log1010 = 1.(b)Find log366:We have 6 =p36 = 3612:
Thus log
366 = log36
361212log3636 =12:Quiz.If log35 = 1465;which of the following numbers is log3004?(a)-2.930(b)-1.465(c)-3.465(d)2.930
Section 6: Use of the Rules of Logarithms 96. Use of the Rules of LogarithmsIn this section we look at some applications of the rules of logarithms.Examples 5(a)log41 = 0:(b)log1010 = 1:(c)log10125 + log108 = log10(1258) = log101000
= log10103= 3log1010 = 3:(d)2log105 + log104 = log1052+ log104 = log10(254)
= log10100 = log10102= 2log1010 = 2:(e)3loga4+loga(1=4)4loga2 = loga43+loga(1=4)loga24
= log a4314loga24= loga42loga24 = log a16loga16 = 0:Section 6: Use of the Rules of Logarithms 10ExerciseUse the rules of logarithms to simplify each of the following.1.3log32log34 + log3122.3log105 + 5log102log1043.2loga6(loga4 + 2loga3)4.5log36(2log34 + log318)5.3log4(p3)12log43 + 3log42log46
Section 7: Quiz on Logarithms 117. Quiz on LogarithmsIn each of the following, ndx:Begin Quiz1.logx1024 = 2(a)23(b)24(c)22(d)252.x= (logap27logap8logap125)=(loga6loga20)(a)1(b)3(c)3/2(d)-2/33.logc(10 +x)logcx= logc5)(a)2.5(b)4.5(c)5.5(d)7.5End Quiz
Section 8: Change of Bases 128. Change of BasesThere is one other rule for logarithms which is extremely useful in
practice. This relates logarithms in one base to logarithms in a dier- ent base. Most calculators will have, as standard, a facility for nding logarithms to the base 10 and also for logarithms to basee(natural logarithms). What happens if a logarithm to a dierent base, forexample 2, is required? The following is the rule that is needed.logac= logablogbc1. Proof of the above rule
Section 8: Change of Bases 13The most frequently used form of the rule is obtained by rearranging the rule on the previous page. We have log ac= logablogbcso logbc=logaclogab:Examples 6(a)Using a calculator we nd that log103 = 047712 and log107 = 084510:Using the above rule,
log37 =log107log103=084510047712= 177124:(b)We can do the same calculation using instead logs to basee.
Using a calculator, log
Levelling-Up Basic MathematicsLogarithmsRobin HoranThe aim of this document is to provide a short, self assessment programme for students who wish to acquire a basic competence in the use of logarithms.Copyrightc2000rhoran@plymouth.ac.ukLast Revision Date: January 16, 2001 Version 1.00
Table of Contents1.Logarithms2.Rules of Logarithms3.Logarithm of a Product4.Logarithm of a Quotient5.Logarithm of a Power6.Use of the Rules of Logarithms7.Quiz on Logarithms8.Change of BasesSolutions to QuizzesSolutions to Problems
Section 1: Logarithms 31. Logarithms (Introduction)LetaandNbe positive real numbers and letN=an:Thennis
called thelogarithm ofNto the basea. We write this asn= logaN:Examples 1(a)Since 16 = 24;then 4 = log216:(b)Since 81 = 34;then 4 = log381:(c)Since 3 =p9 = 912;then 1=2 = log93:(d)Since 31= 1=3;then1 = log3(1=3):
Section 1: Logarithms 4ExerciseUse the denition of logarithm given on the previous page to deter-mine the value ofxin each of the following.1.x= log3272.x= log51253.x= log2(1=4)4.2 = logx(16)5.3 = log2x
Section 2: Rules of Logarithms 52. Rules of LogarithmsLeta;M;Nbe positive real numbers andkbe any number. Then the
following important rules apply to logarithms.1:logaMN= logaM+ logaN2:logaMN= logaMlogaN
3:logamk=klogaM
4:logaa= 1
5:loga1 = 0
Section 3: Logarithm of a Product 63. Logarithm of a Product1. Proof thatlogaMN= logaM+ logaN:Examples 2(a)log64 + log69 = log6(49) = log636:
Ifx= log636;then 6x= 36 = 62:
Thus log
64 + log69 = 2:(b)log520 + log414= log52014:
Now 2014= 5 so log520 + log414= log55 = 1:Quiz.To which of the following numbers does the expression log315 + log306 simplify?(a)4(b)3(c)2(d)1
Section 4: Logarithm of a Quotient 74. Logarithm of a Quotient1. Proof thatlogaMN= logaMlogaN:Examples 3(a)log240log25 = log2405= log28:
Ifx= log28 then 2x= 8 = 23;sox= 3:(b)If log35 = 1:465 then we can nd log306:Since 3=5 = 06;then log306 = log335= log33log35:
Now log
33 = 1;so that log306 = 11465 =0465Quiz.To which of the following numbers does
the expression log212log234simplify?(a)0(b)1(c)2(d)4
Section 5: Logarithm of a Power 85. Logarithm of a Power1. Proof thatlogamk=klogaMExamples 4(a)Find log10(1=10000):We have 10000 = 104;so 1=10000 =
1=104= 104:
Thus log
10(1=10000) = log10104=4log1010 =4;where
we have used rule 4 to write log1010 = 1.(b)Find log366:We have 6 =p36 = 3612:
Thus log
366 = log36
361212log3636 =12:Quiz.If log35 = 1465;which of the following numbers is log3004?(a)-2.930(b)-1.465(c)-3.465(d)2.930
Section 6: Use of the Rules of Logarithms 96. Use of the Rules of LogarithmsIn this section we look at some applications of the rules of logarithms.Examples 5(a)log41 = 0:(b)log1010 = 1:(c)log10125 + log108 = log10(1258) = log101000
= log10103= 3log1010 = 3:(d)2log105 + log104 = log1052+ log104 = log10(254)
= log10100 = log10102= 2log1010 = 2:(e)3loga4+loga(1=4)4loga2 = loga43+loga(1=4)loga24
= log a4314loga24= loga42loga24 = log a16loga16 = 0:Section 6: Use of the Rules of Logarithms 10ExerciseUse the rules of logarithms to simplify each of the following.1.3log32log34 + log3122.3log105 + 5log102log1043.2loga6(loga4 + 2loga3)4.5log36(2log34 + log318)5.3log4(p3)12log43 + 3log42log46
Section 7: Quiz on Logarithms 117. Quiz on LogarithmsIn each of the following, ndx:Begin Quiz1.logx1024 = 2(a)23(b)24(c)22(d)252.x= (logap27logap8logap125)=(loga6loga20)(a)1(b)3(c)3/2(d)-2/33.logc(10 +x)logcx= logc5)(a)2.5(b)4.5(c)5.5(d)7.5End Quiz
Section 8: Change of Bases 128. Change of BasesThere is one other rule for logarithms which is extremely useful in
practice. This relates logarithms in one base to logarithms in a dier- ent base. Most calculators will have, as standard, a facility for nding logarithms to the base 10 and also for logarithms to basee(natural logarithms). What happens if a logarithm to a dierent base, forexample 2, is required? The following is the rule that is needed.logac= logablogbc1. Proof of the above rule
Section 8: Change of Bases 13The most frequently used form of the rule is obtained by rearranging the rule on the previous page. We have log ac= logablogbcso logbc=logaclogab:Examples 6(a)Using a calculator we nd that log103 = 047712 and log