Indices ka matlab hai ek number ko bar bar apne aap se multiply karna. aᵐ mein 'a' base hai aur 'm' index (exponent) hai.
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
(ab)ⁿ = aⁿbⁿ
a⁰ = 1 (a≠0)
a⁻ⁿ = 1/aⁿ
a^(1/n) = ⁿ√a
a^(m/n) = ⁿ√(aᵐ)
Example: Simplify (2³ × 2⁵) / 2⁴
= 2³⁺⁵ / 2⁴ = 2⁸ / 2⁴ = 2⁸⁻⁴ = 2⁴ = 16
📊 Logarithm — Antilogarithm
Logarithm ek inverse operation hai powers ka. Agar aˣ = N, toh log_a(N) = x.
Common Log (base 10) aur Natural Log (base e)
log₁₀(N) = x ⟺ 10ˣ = N [Common log, "log" likhte hain]
loge(N) = x ⟺ eˣ = N [Natural log, "ln" likhte hain, e ≈ 2.718]
Fundamental Laws of Logarithm
Law
Formula
Example
Product Rule
log(MN) = log M + log N
log(100×10) = log 100 + log 10 = 2+1 = 3
Quotient Rule
log(M/N) = log M − log N
log(1000/10) = 3−1 = 2
Power Rule
log(Mⁿ) = n·log M
log(10³) = 3·log 10 = 3
Base Change
log_b(N) = log N / log b
log₂(8) = log 8 / log 2 = 3
Special
log_a(a) = 1, log_a(1) = 0
log 10 = 1, log 1 = 0
Antilogarithm
Agar log N = x, toh N = antilog(x) = 10ˣ. Log table use karke antilog find karte hain.
log N = x → N = antilog(x) = 10ˣ
Example: log N = 2.3010 → N = antilog(2.3010) ≈ 200
(kyunki log 200 = log(2×100) = log 2 + log 100 = 0.3010 + 2 = 2.3010)
Applications of Logarithm
Example 1: Compound Interest using Log
Find time n for ₹1000 to double at 10% CI.
2000 = 1000(1.1)ⁿ → 2 = (1.1)ⁿ
Taking log: log 2 = n·log(1.1)
0.3010 = n × 0.04139
n = 0.3010/0.04139 ≈ 7.27 years