Logarithms

Overview

Engineering reference data for Logarithms in mathematics.

Key Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Symbol	Description	Unit
$\pi$	Pi	3.14159...
$e$	Euler's number	2.71828...

Additional Formulas

Conversion Between Bases

$$\log_a(x) = \frac{\log_c(x)}{\log_c(a)} \tag{10}$$

Conversion Rules (Common & Natural Logarithms)

$$\lg(x) = 0.434294 \cdot \ln(x) \tag{12}$$ $$\ln(x) = 2.302585 \cdot \lg(x) \tag{13}$$ $$\log_2(x) = 1.442695 \cdot \ln(x) = 3.321928 \cdot \lg(x) \tag{15}$$

Properties

• Domain: The logarithm $\log_a(x)$ is only defined for $x > 0$. $\log_a(0)$ and $\log_a(x < 0)$ are undefined.
• Behavior:
  • $\log_a(1) = 0$
  • $\log_a(a) = 1$
  • As $x \to \infty$, $\log_a(x) \to \infty$ (for $a > 1$).

References

• Original Source