Base e and Natural Logarithms 10.5 Notes # ___ Base e and Natural Logarithms 10.5

History The number e is a famous irrational number, and is one of the most important numbers in mathematics. The first few digits are 2.7182818284590452353602874713527... It is often called Euler's number after Leonhard Euler. e is the base of the natural logarithms (invented by John Napier).

The value of (1 + 1/n)n approaches e as n gets bigger and bigger: n Calculating The value of (1 + 1/n)n approaches e as n gets bigger and bigger: n (1 + 1/n)n 1 2.00000 2 2.25000 5 2.48832 10 2.59374 100 2.70481 1,000 2.71692 10,000 2.71815 100,000 2.71827

Vocabulary natural base: the number e, which is found using It is the base rate of growth shared by all continually growing processes natural base exponential function: an exponential function with base e

Use a calculator to estimate to four decimal places. Ex 1 Use a calculator to estimate to four decimal places. Ex 2 Use a calculator to estimate to four decimal places.

Vocabulary natural logarithm: a logarithm with base e The natural log gives you the time needed to reach a certain level of growth. natural logarithmic function: the inverse of the natural base exponential function

Use a calculator to estimate to four decimal places. Ex 3 Use a calculator to estimate to four decimal places. Ex 4 Use a calculator to estimate to four decimal places.

Writing Equivalent Expressions Write an equivalent logarithmic equation. Ex 6 Write an equivalent logarithmic equation.

Writing Equivalent Expressions Write an equivalent exponential equation. Ex 8 Write an equivalent exponential equation.

Inverse Properties

Writing Equivalent Expressions Evaluate Evaluate Ex 11 Ex 12 Evaluate Evaluate

Solving Equations Ex 13 Solve

Solving Equations Ex 14 Solve

Solving Equations Ex 15 Solve

Solving Equations Ex 16 Solve

Solving Inequalities Ex 17 Solve

Solving Inequalities Ex 18 Solve