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Diketahui :
A = ⅙(²log 3³ - ²log 6³ - ²log 12³ + ²log 24³)
Maka nilai 2[tex]^{A}[/tex] adalah ..?
Pembahasan :
Diketahui :
- A = ⅙(²log 3³ - ²log 6³ - ²log 12³ + ²log 24³)
Ditanya :
- 2^A= ... ?
Penyelesaian :
2^A
= 2^⅙(²log 3³ - ²log 6³ - ²log 12³ + ²log 24³)
= 2^⅙(²log 3³ + ²log 24³ - ²log 6³ + ²log 12³ )
[tex] = 2 {}^{ \frac{1}{6} \times log_{2}( \frac{3 {}^{3} \times 24 {}^{3} }{6 {}^{3} \times 12 {}^{3} }) } [/tex]
[tex] = 2 {}^{ \frac{1}{6} \times log_{2} \:( \frac{3 {}^{3} \times ( 3\times 8) {}^{3} }{(2 \times 3{}^{3} )\times (4 \times 3){}^{3} }) } [/tex]
[tex] = 2 {}^{ \frac{1}{6} \times log_{2} \: ( \frac{3 {}^{3} \times 3 {}^{3} \times 8 {}^{3} }{2 {}^{3} \times 3 {}^{3} \times 2 {}^{6} \times 3 {}^{3} } )} [/tex]
[tex] = 2 {}^{ \frac{1}{6} \times log_{2}( \frac{2 {}^{9} }{2 {}^{3} \times 2 {}^{6} } ) } [/tex]
[tex] = 2 {}^{ \frac{1}{6} \times log_{2}(1) } [/tex]
[tex] = 2 {}^{ \frac{1}{6} \times 0} = 2 {}^{0} = 1[/tex]
Menentukan nilai A:
[tex] \rm \: A = \frac{1}{6} \left( {}^{2} log \: {3}^{3} - {}^{2}log \: {6}^{3} - {}^{2}log {12}^{3} + {}^{2}log \: {24}^{3} \right)[/tex]
[tex] \rm \: A = \frac{1}{6} \left( {}^{2} log \: ( \frac{3}{6} ) ^{3} + {}^{2}log \: ( \frac{24}{12} ) ^{3} \right)[/tex]
[tex] \rm \: A = \frac{1}{6} \left( {}^{2} log \: ( \frac{1}{2} ) ^{3} + {}^{2}log \: ( 2 ) ^{3} \right)[/tex]
[tex] \rm \: A = \frac{1}{6} \left( {}^{2} log \: ( {2}^{ - 1} ) ^{3} + {}^{2}log \: ( 2 ) ^{3} \right)[/tex]
[tex] \rm \: A = \frac{1}{6} \left( {}^{2} log \: ( {2} ) ^{ - 3} + 3 \right)[/tex]
[tex] \rm \: A = \frac{1}{6} \left( { - 3} + 3 \right)[/tex]
[tex] \rm \: A = \frac{1}{6} \left( 0 \right)[/tex]
[tex] \rm \: A = 0[/tex]
....
Maka,
2[tex]^{A}[/tex]
= 2[tex]^{0}[/tex]
= 1
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