#A06 · Algebra & Number Theory · Lv.2View in 3D Mind-Map

Logarithm Laws & Change of Base

John Napier · 1614 CE

\log_a(xy) = \log_a x + \log_a y,\quad \log_a b = \frac{\ln b}{\ln a}

a^y = x \iff y = \log_a x \quad (a>0, a\neq 1, x>0)

The inverse of exponentiation that transforms multiplication into addition, powering binary search O(log N), decibel scales, and AI cross-entropy loss.

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Base a

Input x

\log_{2}(64) = \frac{\ln(64)}{\ln(2)} = 6.0000

log_2(x)

6.0000

자연로그 ln(x)

4.1589

상용로그 log₁₀(x)

1.8062

Prerequisite Path (3 STEP)

Full Path Analyzer

Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 2#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 3#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

🚀 Next Unlocks (Derived Formulas) (2)

#C01Limits, Continuity & Euler's Number e #P09Shannon Entropy & Cross-Entropy Loss

Geometric & Intuitive Insight

10을 몇 번 거듭제곱해야 1,000이 될까요? 정답인 3이 바로 log₁₀(1000)입니다. 확률 0.0001 × 0.0001 을 그냥 곱하면 컴퓨터에서 0으로 사라지지만, 로그를 씌우면 -4 + (-4) = -8 로 안전하게 더해집니다.

Step-by-Step Derivation

Step 1.1단계: m = log_a(x), n = log_a(y) 라 두면 정의에 의해 x = a^m, y = a^n 입니다.
Step 2.2단계: 두 식을 곱하면 xy = a^m · a^n = a^(m+n) 이 됩니다.
Step 3.3단계: 다시 로그 정의를 적용하면 log_a(xy) = m + n = log_a(x) + log_a(y) 가 증명됩니다.

AI & Coding Real-World Use

Binary search O(log₂ N) complexity and preventing numerical underflow via Log-Likelihood in AI

#O(log N) Complexity#Log-Likelihood AI#Information Theory

const logBase = (a: number, x: number) => Math.log(x) / Math.log(a);

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