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LICZNIK: log√128+ log32^(⅓)= log128^(½)+ log32^(⅓)=
{128= 2*64= 2*8*8= 2*2³*2³=2⁷, 32=2*16=2*2⁴= 2⁵}
=log(2⁷)^(½)+ log(2⁵)^(⅓)= log2^(⁷/₂)+ log2^(⁵/₃)=
{potęgowanie potęgi (2⁷)^(½)= 2^(⁷*½)= 2^(⁷/₂)
=log(2⁵)^(⅓)= 2^(⁵/₃)}
=log[2^(⁷/₂)*2^(⁵/₃)]= log2^(³¹/₆)= ³¹/₆log2
{korzystamy z własności logarytmów loga+ logb= log(a*b)}
{zapis 128^(½) sto dwadzieścia osiem do potęgi ½}
MIANOWNIK: log(2√2)= log[2*2^(½)]= log2^(³/₂)= ³/₂log2
Mamy licznik/mianownik:
[³¹/₆log2]/[³/₂log2]= ³¹/₆ : ³/₂= ³¹/₆ * ²/₃= ³¹/₉= 3⁴/₉
Zad. 20
mamy log12= a, log2= b
obliczamy log 90= log[(90*16)/16]= log(90*16)- log16=
log(9*16*10)- log2⁴= log(144*10)- log2⁴= log144+ log10- log2⁴=
log12²+ log10- log2⁴= 2log12+ log10- 4log2=
2a+ 1- 4b= 2a- 4b+ 1
{log10= 1, bo 10¹= 10
{liczbę 90 możemy rozłożyć:
90= (90*16)/16= (9*16*10)/16= (144*10)/16= (12²*10)/(2⁴)}