Pertanyaan

Tentukan nilai 2 p yang memenuhi: ( p 2 + 4 ) lo g ( p + 1 ) = 3 lo g ( 5 ) ⋅ 2 lo g ( 81 ) 2 lo g ( 5 )

Tentukan nilai $2 p$ yang memenuhi:

$^{_{(p^{2} + 4)}} lo g (p + 1) = \frac{^{2} lo g ( 5 )}{^{3} lo g ( 5 ) \cdot ^{2} lo g ( 81 )}$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

nilai 2 p yang memenuhi logaritma tersebtu adalah 3 .

nilai

2 p

yang memenuhi logaritma tersebtu adalah

3

Pembahasan

Ingat kembali: a lo g b ⋅ b lo g c = a lo g c a lo g a = 1 a lo g b m = m ⋅ a lo g b Maka: p 2 + 4 lo g ( p + 1 ) p 2 + 4 l o g p + 1 p 2 + 4 lo g p + 1 p 2 + 4 lo g p + 1 p 2 + 4 lo g p + 1 p 2 + 4 lo g p + 1 p 2 + 4 lo g p + 1 = = = = = = = 3 l o g ( 5 ) ⋅ 2 l o g ( 81 ) 2 l o g ( 5 ) 3 l o g 5 2 1 ⋅ 2 l o g 3 4 2 l o g 5 2 1 ⋅ 3 l o g 5 ⋅ 4 ⋅ 2 l o g 3 2 l o g 5 2 1 ⋅ 4 ⋅ 2 lo g 3 ⋅ 3 lo g 5 2 l o g 5 2 ⋅ 2 lo g 5 2 l o g 5 2 1 2 1 ⋅ p 2 + 4 lo g p 2 + 4 Ingatpersamaan logaritma: f ( x ) lo g g ( x ) = f ( x ) lo g h ( x ) maka g ( x ) = h ( x ) ,dengan f ( x ) > 0 , f ( x )  = 1 , dan g ( x ) , h ( x ) > 0 . Sehingga: Karena f ( x ) > 0 , f ( x )  = 1 , dan g ( x ) , h ( x ) > 0 , maka: p 2 + 4 = = = ( 2 3 ) 2 + 4 4 9 + 4 4 4 13 p + 1 = = 2 3 + 1 2 5 Diperoleh f ( x ) > 0 dan f ( x )  = 1 , g ( x ) > 0 , dan h ( x ) > 0 , sehingga nilai x = 2 3 memenuhi persamaan tersebut. 2 p = 2 ( 2 3 ) = 3 Jadi,nilai 2 p yang memenuhi logaritma tersebtu adalah 3 .

Ingat kembali:

$^{a} lo g b \cdot^{b} lo g c =^{a} lo g c^{a} lo g a = 1^{a} lo g b^{m} = m \cdot^{a} lo g b$

Maka:

$^{p^{2} + 4} lo g (p + 1)^{p^{2} + 4} l o g p + 1^{p^{2} + 4} lo g p + 1^{p^{2} + 4} lo g p + 1^{p^{2} + 4} lo g p + 1^{p^{2} + 4} lo g p + 1^{p^{2} + 4} lo g p + 1 = = = = = = = \frac{^{2} l o g ( 5 )}{^{3} l o g ( 5 ) \cdot ^{2} l o g ( 81 )} \frac{^{2} l o g 5}{^{3} l o g 5 ^{\frac{1}{2}} \cdot ^{2} l o g 3 ^{4}} \frac{^{2} l o g 5}{\frac{1}{2} \cdot ^{3} l o g 5 \cdot 4 \cdot ^{2} l o g 3} \frac{^{2} l o g 5}{\frac{1}{2} \cdot 4 \cdot ^{2} lo g 3 \cdot ^{3} lo g 5} \frac{^{2} l o g 5}{2 \cdot ^{2} lo g 5} \frac{1}{2} \frac{1}{2} \cdot^{p^{2} + 4} lo g p^{2} + 4$

Ingat persamaan logaritma:

$^{f (x)} lo g g (x) =^{f (x)} lo g h (x)$

maka $g (x) = h (x)$ , dengan $f (x) > 0, f (x) \neq = 1, dan g (x), h (x) > 0$ .

Sehingga:

$table attributes columnalign right center left columnspacing 0px end attributes row cell p plus 1 end cell equals cell open parentheses p squared plus 4 close parentheses to the power of 1 half end exponent end cell row cell open parentheses p plus 1 close parentheses squared end cell equals cell open parentheses open parentheses p squared plus 4 close parentheses to the power of fraction numerator 1 over denominator up diagonal strike 2 end fraction end exponent close parentheses to the power of up diagonal strike 2 end exponent end cell row cell p squared plus 2 p plus 1 end cell equals cell p squared plus 4 end cell row cell 2 p plus 1 end cell equals 4 row cell 2 p end cell equals cell 4 minus 1 end cell row cell 2 p end cell equals 3 row p equals cell 3 over 2 end cell end table$

Karena $f (x) > 0, f (x) \neq = 1, dan g (x), h (x) > 0$ , maka: