Essential Logarithm Formulas

Basic Properties

Definition

For any positive numbers $M$, $b$ (where $b ≠ 1$):

$y = \log_b(M) \iff b^y = M$

Basic Values

Expression	Value	Reason
$\log_b(1)$	$0$	$b^0 = 1$
$\log_b(b)$	$1$	$b^1 = b$
$\log_b(b^n)$	$n$	$b^n = b^n$

Core Rules

Product Rule

When multiplying numbers inside a logarithm:

$\log_b(M × N) = \log_b(M) + \log_b(N)$

Quotient Rule

When dividing numbers inside a logarithm:

$\log_b(M ÷ N) = \log_b(M) - \log_b(N)$

Power Rule

When dealing with exponents:

$\log_b(M^n) = n × \log_b(M)$

Change of Base

General Formula

To convert between different bases:

$\log_b(M) = \frac{\log_a(M)}{\log_a(b)}$

Where $a$ can be any base (commonly 10 or e)

Common Conversions

From	To	Formula
$\ln(x)$	$\log_{10}(x)$	$\ln(x) × 0.4343$
$\log_{10}(x)$	$\ln(x)$	$\log_{10}(x) × 2.303$

Common Patterns

Exponential Equations

To solve $b^x = M$:

$x = \log_b(M)$

Example: If $2^x = 8$, then $x = \log_2(8) = 3$

Logarithmic Equations

To solve $\log_b(x) = M$:

$x = b^M$

Example: If $\log_{10}(x) = 2$, then $x = 10^2 = 100$

Quick Reference Table

Rule	Formula
Product	$\log_b(M × N) = \log_b(M) + \log_b(N)$
Quotient	$\log_b(M ÷ N) = \log_b(M) - \log_b(N)$
Power	$\log_b(M^n) = n × \log_b(M)$
Change of Base	$\log_b(M) = \frac{\log_a(M)}{\log_a(b)}$
Definition	$\log_b(M) = y \iff b^y = M$

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