Solving A Logarithmic System

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Комментарии
Автор

У нас десятичный логарифм обозначается Lg, также как натуральный Ln, а другой Log с обязательным указанием основания.

andrei_nazimov
Автор

X = 10^(3 + sqrt(6)) and y = 10^( 3 - sqrt(6)) and vice versa

kianmath
Автор

Let us see. Assume log means log base 10.
logy*logx=log1000=3; log(xy)=logx + logy =
logy=6–logx; (6–logx)*logx=3; logx^2–6logx+3=0; logx=(6±2√6)/2=3±√6; logy=6–logx=3∓√6
x=281507.35..., y=3.552305... or y=281507.35..., x=3.552305...

roger
Автор

it seems very easy.. let's see..

logy.logx = 3
logx + logy = 6

logx = a; logy = b

a.b = 3
a + b = 6 => b = 6 - a

a.(6 - a) = 3
a² - 6a + 3 = 0
a = (6 ± 2√6)/2 => a = 3 ± √6

a = 3 + √6 => b = 3 - √6
x = 10^(3 + √6)
y = 10^(3 - √6)

a = 3 - √6 => b = 3 + √6
x = 10^(3 - √6)
y = 10^(3 + √6)

let's proof
x.y = 10^(3 + √6 + 3 - √6) = 10^6
x^logy = 10^[(3 + √6)(3 - √6)] = 10^3

SidneiMV