Base e and Natural Logarithms Section 10-5

Base e ●A●A●A●As n i i i increases the expression approaches the irrational number … ●●T●●This is the natural base or e ●s●s●s●sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant Euler: He was also a student of Daniel Bernoulli

e calculated to 100,000 digits v At font size 3

Base e shows up in problems that involve the statistical bell curve There is also a connection between e and via Euler's Equation. complex numbers, This remarkable equation involves the five most important constants in all of mathematics: 0, 1, i, and e.

Natural Base Exponential Functions ●A●A●A●An exponential function with base e is a natural base exponential function ●●U●●Used extensively in science to model quantities that grow and decay continuously

Natural Logarithms ● A logarithm with base e is a natural logarithm ● Sometimes written as ● Usually written as the abbreviated form ln x ● y= ln x is called a natural logarithm function

Natural Logarithm natural base exponential functions and natural logarithms are inverses Natural Base Exponential Function

ln x= By using the following equation, you can write an equivalent natural base exponential equation for a natural logarithmic equation and vice versa Write an equivalent exponential or logarithmic equation

Inverse Property of Base e and Natural Logarithms Evaluate each expression

Solving Base e Equations or Inequalities

Compound Interest r is the annual interest rate P is amount of principal

Solve If you deposit $1000 in an account paying 5% annual interest compounded continuously: a. What is the balance after 10 years? b. How long will it take for the balance in your account to reach at least 1500 The balance after ten years would be It will take at least 8.11 years for the balance to reach %1500