04 // Interactive Formula Compendium // CALCULUS-ALGEBRAlog_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)
Logarithm Laws & Change of Base Formula — Complete Algebraic Sheet
The logarithm log_b(x) is the exponent y to which base b > 0 (b ≠ 1) must be raised to produce x > 0 (bʸ = x), and can be evaluated in any base via the change-of-base identity log_b(x) = ln(x) / ln(b).
Formula Reference
Main Formula
log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)
Rearranged Forms
log_b(x · y) = log_b(x) + log_b(y) (Product Rule)
log_b(x / y) = log_b(x) − log_b(y) (Quotient Rule)
log_b(xᵖ) = p · log_b(x), b^(log_b x) = x (Power & Inverse Identity)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| log_b(x) | Logarithm of x in Base b | dimensionless |
| x | Argument Value (x > 0) | 1 |
| b | Logarithm Base (b > 0, b ≠ 1) | 1 |
Inputs & Parameters
[1]
[1]
Calculated Result (Logarithm of x in Base b)
log_b(x) =10dimensionless
Formula Notes & Background
- Let y = log_b(x), which by definition means bʸ = x.
- Take the natural logarithm ln(·) of both sides: ln(bʸ) = ln(x).
- Apply the power property y · ln(b) = ln(x) and divide by ln(b) ≠ 0 to obtain y = ln(x) / ln(b).
Step-by-Step Examples
#1: Evaluate log₂(1024) and log₂(50) using natural logarithms.
x₁ = 1024, b = 2x₂ = 50, b = 2
log₂(1024) = ln(1024)/ln(2) = 10; log₂(50) = 3.91202 / 0.69315
→ log₂(1024) = 10, log₂(50) ≈ 5.64386
#2: Solve for compounding time t in 2 = (1.07)ᵗ (the exact Rule of 72 doubling time).
x = 2b = 1.07
t = log_{1.07}(2) = ln(2) / ln(1.07) = 0.69315 / 0.06766
→ t = 10.2448 years
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Frequently Asked Questions
Why is log_b(x) undefined for x ≤ 0 in real numbers?+
For any positive base b > 0, the exponential function bʸ is strictly positive for all real y, so it never reaches 0 or negative numbers.
What is the derivative of log_b(x) with respect to x?+
Since log_b(x) = ln(x)/ln(b), its derivative is d/dx[log_b(x)] = 1 / (x · ln(b)).
Where do base-2, base-e, and base-10 logarithms appear in practice?+
Base 2 measures information bits and binary tree depth, base e governs continuous calculus and physical decay, and base 10 powers decibel (dB), pH, and Richter scales.