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The Special Number e and the Natural Logarithm

The Special Number e and the Natural Logarithm. Date: The symbol e is an irrational number whose value is defined as To find a power of e, use the _____________ button on your calculator. The inverse function of.

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The Special Number e and the Natural Logarithm

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  1. The Special Number e and the Natural Logarithm • Date: • The symbol e is an irrational number whose value is defined as • To find a power of e, use the _____________ button on your calculator.

  2. The inverse function of • Is given by the function f(x)=ln(x), also called the _________________________, and to find this function of a number on a calculator, the _____________button can be used. • The relationship between the two functions can be shown by:

  3. The following are sketches of the two functions:

  4. Powers of e and natural logarithms of numbers can be estimated and calculated: • Ex. Determine • Ex. Determine ln(82)

  5. Because these two functions are inverses of one another, we can determine two formulae:

  6. The laws of logarithms also apply to the natural logarithm, where x,y > 0: • 1) ln(xy) = lnx + lny • 2)ln(x/y) = lnx – lny • 3)

  7. Examples: • Determine

  8. Exponential and Natural Logarithmic Equations can be solved as well: • Example 1:

  9. Example 2: • Solve for x if

  10. Application: • The population of a bacterial culture as a function of time in days is given by the formula • a) Determine the initial population. • b) Determine the population after one week.

  11. Application: • The population of a bacterial culture as a function of time in days is given by the formula • Determine when the population will be 425% of what it was.

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