Header Ads

Laws of Logarithms

Introduction
math2everThree most important Laws of Logarithms Rules of logarithms are:

  1. Rule of Multiplication
  2. Rule of Division
  3. Rule of Power 


Three important rules of indices are listed in table 1. Rules of multiplication and division are applied for logarithms containing identical base.

No.        Rules of Logarithm Examples
1 Rule of Multiplication
log a XY = log a X  + log a Y MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamiwaiaadMfacaqG9aGaaeiiaiaabYgacaqGVbGaae4zamaaBaaaleaacaqGHbaabeaakiaadIfacaqGGaGaae4kaiaabccacaqGSbGaae4BaiaabEgadaWgaaWcbaGaaeyyaaqabaGccaWGzbaaaa@4B2D@
log a ( 2×3 ) = log a 2  + log a 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqaaaMbciGGSbGaai4BaiaacEgadaWgaaWcbaGaamyyaaqabaGcdaqadaqaaiaaikdacqGHxdaTcaaIZaaacaGLOaGaayzkaaGaaeypaiaabccacaqGSbGaae4BaiaabEgadaWgaaWcbaGaaeyyaaqabaGccaaIYaGaaeiiaiaabUcacaqGGaGaaeiBaiaab+gacaqGNbWaaSbaaSqaaiaabggaaeqaaOGaaG4maaaa@4EB2@
2 Rule of Division
log a X Y = log a X  - log a Y MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOWaaSaaaeaacaWGybaabaGaamywaaaacaqG9aGaaeiiaiaabYgacaqGVbGaae4zamaaBaaaleaacaqGHbaabeaakiaadIfacaqGGaGaaeylaiaabccacaqGSbGaae4BaiaabEgadaWgaaWcbaGaaeyyaaqabaGccaWGzbaaaa@4B3F@
log a ( 4 5 ) = log a 4 - log a 5 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqaaaMbciGGSbGaai4BaiaacEgadaWgaaWcbaGaamyyaaqabaGcdaqadaqaamaalaaabaGaaGinaaqaaiaaiwdaaaaacaGLOaGaayzkaaGaaeypaiaabccacaqGSbGaae4BaiaabEgadaWgaaWcbaGaaeyyaaqabaGccaqG0aGaaeiiaiaab2cacaqGGaGaaeiBaiaab+gacaqGNbWaaSbaaSqaaiaabggaaeqaaOGaaGynaaaa@4CAE@
3 Rule of Power
log a X n   = nlog a X MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamiwamaaCaaaleqabaGaamOBaaaakiaabccacaqGGaGaaeypaiaabccacaqGUbGaaeiBaiaab+gacaqGNbWaaSbaaSqaaiaabggaaeqaaOGaamiwaaaa@46F9@
log a 7 2 =2 log a 7 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqaaaMbciGGSbGaai4BaiaacEgadaWgaaWcbaGaamyyaaqabaGccaqG3aWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaGOmaiGacYgacaGGVbGaai4zamaaBaaaleaacaWGHbaabeaakiaaiEdaaaa@451B@
Table 1
Three other laws of logarithms are stated as below.

No.        Rules of indices Examples
1 Logarithm equal to 1
log a a =1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamyyaiaabccacaqG9aGaaeymaaaa@3F93@
log 5 5=1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqaaaMbciGGSbGaai4BaiaacEgadaWgaaWcbaGaaGynaaqabaGccaqG1aGaeyypa0JaaGymaaaa@3F51@
2 Logarithm equal to 0
log a 1 =0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaaGymaiaabccacaqG9aGaaeimaaaa@3F67@
log 2 1=0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqaaaMbciGGSbGaai4BaiaacEgadaWgaaWcbaGaaGOmaaqabaGccaaIXaGaeyypa0JaaGimaaaa@3F50@
3 log a 1 X  = log a X 1  = -log a X MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOWaaSaaaeaacaaIXaaabaGaamiwaaaacaqGGaGaaeypaiaabccacaqGSbGaae4BaiaabEgadaWgaaWcbaGaaeyyaaqabaGccaWGybWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaeiiaiaab2dacaqGGaGaaeylaiaabYgacaqGVbGaae4zamaaBaaaleaacaqGHbaabeaakiaadIfaaaa@4E5D@ log 2 1 7 = log 2 7 1 = log 2 7 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiGaaGabaaMbciGGSbGaai4BaiaacEgadaWgaaWcbaGaaGOmaaqabaGcdaWcaaqaaiaaigdaaeaacaaI3aaaaiabg2da9iGacYgacaGGVbGaai4zamaaBaaaleaacaaIYaaabeaakiaaiEdadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGH9aqpcqGHsislciGGSbGaai4BaiaacEgadaWgaaWcbaGaaGOmaaqabaGccaaI3aaaaa@4C4C@
Table 2
Changing the base of logarithm
  • Consider the following logarithm with base a and law used to change the base to base c. 
  • The new base is c with two new logarithms are in division operation.
From base a To base c
log a b MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamOyaaaa@3D7D@ log a b= log c b log c a MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiGaaGabaaMbciGGSbGaai4BaiaacEgadaWgaaWcbaGaamyyaaqabaGccaWGIbGaeyypa0ZaaSaaaeaaciGGSbGaai4BaiaacEgadaWgaaWcbaGaam4yaaqabaGccaWGIbaabaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadogaaeqaaOGaamyyaaaaaaa@48B2@
Table 2
Powered by Blogger.