Question

Kuiss Rumit Aljabar

Jika [tex]\rm{}^{x} {\log}^{2}y - {}^{x} \log {y}^{5} = - 1[/tex]

Maka, nilai dari [tex]\rm{}^{x} \log {y}^{5} +{ ({}^{x} \log y)}^{ - 2} = \dots[/tex]

a. 12
b. 24
c. tidak terdefinisi
d. 10

#Hari ke-26
​

Answer 1

Verified answer

Nilai logaritma [tex]^x \: log \: y^5 \:+\: (^x \: log \: y)^{- 2}[/tex] adalah 24. Jawaban B. Menggunakan pemisalan dan sifat-sifat logaritma.

Penjelasan dengan langkah-langkah:

Diketahui:

• [tex]^x \: log^2 \: y \:-\: ^x \: log \: y^5 \:=\: - 1[/tex]

Ditanyakan:

• [tex]^x \: log \: y^5 \:+\: (^x \: log \: y)^{- 2}[/tex]?

Jawaban:

Misalkan [tex]^x \: log \: y[/tex] = p maka

[tex]^x \: log^2 \: y \:-\: ^x \: log \: y^5 \:=\: - 1[/tex]

[tex](^x \: log \: y)^2 \:-\: 5 \: ^x \: log \: y \:=\: - 1[/tex]

[tex]p^2 \:-\: 5p \:=\: - 1[/tex]

[tex]p^2 \:-\: 5p \:+\: 1 \:=\: 0[/tex]

• a = 1
• b = - 5
• c = 1

Menggunakan rumus ABC untuk persamaan kuadrat ap² + bp + c = 0

[tex]p_{1 , 2} \:=\: \frac{- b \pm \sqrt{b^2 \:-\: 4ac}}{2a}[/tex]

[tex]p_{1 , 2} \:=\: \frac{5 \pm \sqrt{(- 5)^2 \:-\: (4 \times 1 \times 1)}}{2 \times 1}[/tex]

[tex]p_{1 , 2} \:=\: \frac{5 \pm \sqrt{25 \:-\: 4}}{2}[/tex]

[tex]p_{1 , 2} \:=\: \frac{5 \pm \sqrt{21}}{2}[/tex]

• [tex]p_1 \:=\: \frac{5 \:+\: \sqrt{21}}{2}[/tex]
• [tex]p_2 \:=\: \frac{5 \:-\: \sqrt{21}}{2}[/tex]

[tex]^x \: log \: y^5 \:+\: (^x \: log \: y)^{- 2}[/tex]

= [tex]5 \: ^x \: log \: y \:+\: (\frac{1}{^x \: log \: y})^2[/tex]

= [tex]5p \:+\: \frac{1}{p^2}[/tex]

Untuk p₁

[tex]5p_1 \:+\: \frac{1}{p_1^2}[/tex]

= [tex]5 \times \frac{5 \:+\: \sqrt{21}}{2} \:+\: \frac{1}{(\frac{5 \:+\: \sqrt{21}}{2})^2}[/tex]

= [tex]\frac{25 \:+\: 5 \sqrt{21}}{2} \:+\: \frac{1}{(\frac{25 \:+\: 10 \sqrt{21} \:+\: 21}{4})}[/tex]

= [tex]\frac{25 \:+\: 5 \sqrt{21}}{2} \:+\: \frac{4}{46 \:+\: 10 \sqrt{21}}[/tex]

= [tex]\frac{25 \:+\: 5 \sqrt{21}}{2} \:+\: \frac{4}{2 \: (23 \:+\: 5 \sqrt{21})}[/tex]

= [tex]\frac{25 \:+\: 5 \sqrt{21}}{2} \:+\: \frac{2}{23 \:+\: 5 \sqrt{21}}[/tex]

= [tex]\frac{25 \:+\: 5 \sqrt{21}}{2} \:+\: \frac{2}{23 \:+\: 5 \sqrt{21}} \times \frac{23 \:-\: 5 \sqrt{21}}{23 \:-\: 5 \sqrt{21}}[/tex]

= [tex]\frac{25 \:+\: 5 \sqrt{21}}{2} \:+\: \frac{2 \: (23 \:-\: 5 \sqrt{21})}{529 \:-\: 525}[/tex]

= [tex]\frac{25 \:+\: 5 \sqrt{21}}{2} \:+\: \frac{2 \: (23 \:-\: 5 \sqrt{21})}{4}[/tex]

= [tex]\frac{25 \:+\: 5 \sqrt{21}}{2} \:+\: \frac{23 \:-\: 5 \sqrt{21}}{2}[/tex]

= [tex]\frac{25 \:+\: 5 \sqrt{21} \:+\: 23 \:-\: 5 \sqrt{21}}{2}[/tex]

= [tex]\frac{48}{2}[/tex]

= 24

Untuk p₂

[tex]5p_2 \:+\: \frac{1}{p_2^2}[/tex]

= [tex]5 \times \frac{5 \:-\: \sqrt{21}}{2} \:+\: \frac{1}{(\frac{5 \:-\: \sqrt{21}}{2})^2}[/tex]

= [tex]\frac{25 \:-\: 5 \sqrt{21}}{2} \:+\: \frac{1}{(\frac{25 \:-\: 10 \sqrt{21} \:+\: 21}{4})}[/tex]

= [tex]\frac{25 \:-\: 5 \sqrt{21}}{2} \:+\: \frac{4}{46 \:-\: 10 \sqrt{21}}[/tex]

= [tex]\frac{25 \:-\: 5 \sqrt{21}}{2} \:+\: \frac{4}{2 \: (23 \:-\: 5 \sqrt{21})}[/tex]

= [tex]\frac{25 \:-\: 5 \sqrt{21}}{2} \:+\: \frac{2}{23 \:-\: 5 \sqrt{21}}[/tex]

= [tex]\frac{25 \:-\: 5 \sqrt{21}}{2} \:+\: \frac{2}{23 \:-\: 5 \sqrt{21}} \times \frac{23 \:+\: 5 \sqrt{21}}{23 \:+\: 5 \sqrt{21}}[/tex]

= [tex]\frac{25 \:-\: 5 \sqrt{21}}{2} \:+\: \frac{2 \: (23 \:+\: 5 \sqrt{21})}{529 \:-\: 525}[/tex]

= [tex]\frac{25 \:-\: 5 \sqrt{21}}{2} \:+\: \frac{2 \: (23 \:+\: 5 \sqrt{21})}{4}[/tex]

= [tex]\frac{25 \:-\: 5 \sqrt{21}}{2} \:+\: \frac{23 \:+\: 5 \sqrt{21}}{2}[/tex]

= [tex]\frac{25 \:-\: 5 \sqrt{21} \:+\: 23 \:+\: 5 \sqrt{21}}{2}[/tex]

= [tex]\frac{48}{2}[/tex]

= 24

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