Natural Logarithm

Natural logarithm, abbreviated as, \(ln\), also called natural log, is a mathematical function that describes how quantities grow or decay in processes that follow continuous, exponential behavior.  It is written as \(ln (x)\) and is defined as the logarithm to the base \(e\), is an important mathematical constant approximately equal to \(2.71828\).  The natural logarithm tells you the power to which emust be raised to produce a given number \(x\).  Because many natural processes, such as population growth, radioactive decay, heat transfer, and compound interest, change continuously rather than in steps, the natural logarithm appears frequently in science, engineering, and calculus.  It also has a strong geometric meaning: ln(x) represents the area under the curve 1/t from 1 to x, linking it closely with integration and continuous change. 

Natural Logarithm Rules

Power Rule

• \( ln \left( x^y \right) = y \left[ ln \left( x \right) \right] \)

Product Rule  -  

• \( ln \left( x \right)\left( x \right) = ln \left( x \right) + ln \left( y \right)  \)

Quotient Rule

• \( ln \left( \frac{x}{y} \right) = ln \left( x \right) - ln \left( y \right)  \)

Reciprocal Rule

• \( ln \left( \frac{1}{x} \right) = ln \left( x \right)  \)

Natural Logarithm Properties

• For between 0 and 1
  • As x nears 0, it heads to infinity
  • As x increases it heads to - infinity
  • It is a strictly decreasing function
  • It has a vertical asymptote along the y-axis (x=0)
   
• For a above 1
  • As x nears 0, it heads to - infinity
  • As x increases it heads to infinity
  • It is a strictly decreasing function
  • It has a vertical asymptote along the y-axis (x=0)