Natural log: ln

Natural log: ln

Lessons

• Definition of `` natural logarithm"" and mathematical constant ``e":":

1)
Recall: common logarithms = log with base 10"``10"
example:log3=log103 \log3 = \log_{10}3
natural logarithms = log with base ``e""
example:ln5=loge5 \ln5 = \log_e5


2)
Like π"``\pi" , a mathematical constant equal to 3.14….., `` e"" is just another mathematical constant equal to 2.71…. .


3)
Significance of π"``\pi" : we use it in circle calculations:
example: areacircle=πr2 area_{circle} = \pi r^2
or
circumferencecircle=2πr circumference_{circle} = 2 \pi r
Significance of ``e"" : we use it mostly in calculus. `` e"" is a unique number such that the slope of tangent line at every point on the graph of f(x)=ex f(x) = e^x is equal to the y-value of the point.

  • 1.
    Evaluate ln5
    a)
    by using the LOG key on a calculator.

    b)
    by using the LN key on a calculator.


  • 2.
    Without using a calculator, evaluate:
    a)
    lne\ln e
    [useful rule:lne=1] \ln e = 1]

    b)
    eln500 e^{\ln500}
    [useful rule: elna=a] e^{\ln a} = a]