7. $$ 2^{x^{2}-3} \cdot 5^{x^{2}-3}=0,01\left(10^{x-1}\right)^{3} $$

Question

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Berilgan tenglama:
$$ 2^{x^{2}-3} \cdot 5^{x^{2}-3} = 0,01 \left(10^{x-1}\right)^{3} $$

**Qadam 1: Tenglamaning ikki tomonini soddalashtirish**

Chap tomonni soddalashtiramiz:
$$ 2^{x^{2}-3} \cdot 5^{x^{2}-3} = (2 \cdot 5)^{x^{2}-3} = 10^{x^{2}-3} $$

O'ng tomonni soddalashtiramiz:
$$ 0,01 \left(10^{x-1}\right)^{3} = 0,01 \cdot 10^{3(x-1)} = 0,01 \cdot 10^{3x -3} $$

$$ 0,01 = 10^{-2} $$, shuning uchun:
$$ 10^{-2} \cdot 10^{3x -3} = 10^{3x -5} $$

Endi tenglama quyidagicha ko'rinadi:
$$ 10^{x^{2}-3} = 10^{3x -5} $$

**Qadam 2: Eksponent darajalar tengligini yechish**

Agar $$10^a = 10^b$$ bo'lsa, unda $$a = b$$. Shunday ekan:
$$ x^{2} - 3 = 3x -5 $$

**Qadam 3: Kvadrat tenglamani yechish**

Tenglamani tartibga solamiz:
$$ x^{2} - 3x + 2 = 0 $$

Bu kvadrat tenglama yechimlarini topamiz:
$$ x = \frac{3 \pm \sqrt{9 - 8}}{2} = \frac{3 \pm 1}{2} $$

Demak, yechimlar:
$$ x = \frac{3 + 1}{2} = 2 $$
$$ x = \frac{3 - 1}{2} = 1 $$

**Natija:**
$$ x = 1 \quad \text{yoki} \quad x = 2 $$
Natija: $$ x = 1 $$ yoki $$ x = 2 $$.

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