Logarifm nima? (8-sinf matematika uchun tushunarli va to‘liq darslik)

DO`STLARGA ULASHING:

Logarifm tushunchasi

Logarifm — bu daraja (eksponenta) ning teskarisi bo‘lgan matematik amal.

Agar:

ax=ba^x = bax=b

bo‘lsa, u holda:

log⁡ab=x\log_a b = xloga​b=x

👉 Ya’ni, logarifm — “qaysi darajaga ko‘tarilganda a sonidan b hosil bo‘ladi?” degan savolga javob beradi.

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Oddiy misollar

• 102=10010^2 = 100102=100 → log⁡10100=2\log_{10}100 = 2log10​100=2

• 23=82^3 = 823=8 → log⁡28=3\log_2 8 = 3log2​8=3

• 51=55^1 = 551=5 → log⁡55=1\log_5 5 = 1log5​5=1

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Logarifmning asosiy qismlari

log⁡ab\log_a bloga​b

• a — logarifm asosi (a > 0, a ≠ 1)

• b — logarifm ostidagi son (b > 0)

• natija — daraja

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Eng ko‘p ishlatiladigan logarifmlar

1. O‘nli logarifm

   log⁡10x=log⁡x\log_{10} x = \log xlog10​x=logx

   (asosi 10 bo‘lgan logarifm)

2. Natural logarifm (keyingi sinflarda)

   $$\ln x$$

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Logarifmning asosiy xossalari

1. log⁡a(xy)=log⁡ax+log⁡ay\log_a (xy) = \log_a x + \log_a yloga​(xy)=loga​x+loga​y

2. log⁡a(xy)=log⁡ax−log⁡ay\log_a \left(\frac{x}{y}\right) = \log_a x — \log_a yloga​(yx​)=loga​x−loga​y

3. log⁡axn=nlog⁡ax\log_a x^n = n \log_a xloga​xn=nloga​x

4. log⁡aa=1\log_a a = 1loga​a=1

5. log⁡a1=0\log_a 1 = 0loga​1=0

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Masala (imtihon tipida)

Masala:
log⁡216\log_{2} 16log2​16 ni toping.

Yechish:
24=162^4 = 1624=16
Demak:

log⁡216=4\log_{2} 16 = 4log2​16=4

✅ Javob: 4

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Logarifm qayerda ishlatiladi?

• Fizikada (tovush balandligi — detsibel)

• Kimyoda (pH shkalasi)

• Informatikada

• Astronomiyada

• Iqtisod va statistika hisoblarida