Question

`Evaluate \: the \: following \\ (1) \: log1000 \: \\ (2) ( (128)/(625) ) \\ (3) log {x}^(2) {y}^(3) {z}^(4) \\ (4) log \frac{ {p}^(2) {q}^(3) }{r} \\ (5) log \sqrt{ \frac{ {x}^(3) }{ {y}^(2) } }`


`If \: {x}^(2) + {y}^(2) = 25xy. \\ Then \: prove \: that \: \\ 2 log(x + y) = \\ 3 log3 + logx + logy. \:`

`\: \: \: \: \: \: \: \: \: \: \: \: \:I \: need \: the \: answer \\ \: \: \: \: \: \: \: \: \: \: \: \: Plz \: fastly \:`
I need ans !!!!!
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Answer 1

Explanation:

1. log1000= log 10³= 3 log10 =3
2. log(128/625)= 7 log 2+ 4 log 5
3. log x²y³z⁴= 2 logx + 3 log y + 4 log z
4. log p²q³/r= 2 log p +3 log q - log r
5. log√(x³/y²)=3/2[ log (x)] - log y

x²+y²=25xy

(x+y)²-2xy=25xy

(x+y)²= 2xy +25 xy

=27xy

Take log on both sides

2 log(x+y) =log 27 + log x + log y

=log 3³+ log x + log y

2 log(x+y)=3 log 3 + log x + log y

Answer 2

(1)

• log 1000 = log 10³ = 3 log 10 = 3

(2) It should be with log? If yes ignore log x and consider the right side

• (128/625) = x
• log x = log (128/625)
• log x = log 128 - log 625
• log x = 7 log 2 - 4 log 5

(3)

• log (x²y³z⁴) = log x² + log y³ + log z⁴ = 2 log x + 3 log y + 4 log z

(4)

• log (p²q³/r) = log p² + log q³ - log r = 2 log p + 3 log q - log r

(5)

• log
  `\sqrt{(x^3)/(y2) }` = 1/2 log
  `x^3y^(-2)` = 3/2 log x - log y

(6)

• x² + y² = 25xy
• x² + 2xy + y² = 27xy
• (x + y)² = 27xy
• log (x + y)² = log (27xy)
• 2 log (x + y) = log 3³+ log x + log y
• 2 log (x + y) = 3 log 3 + log x + log y
• Proved