# Natural log: ln ##### Examples
###### Lessons
1. Evaluate ln5
1. by using the LOG key on a calculator.
2. by using the LN key on a calculator.
2. Without using a calculator, evaluate:
1. $\ln e$
[useful rule:$\ln e = 1]$
2. $e^{\ln500}$
[useful rule: $e^{\ln a} = a]$
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###### Topic Notes
• Definition of $$natural logarithm$"$and mathematical constant $$e$":$

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

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

3)
Significance of $\pi"$: we use it in circle calculations:
example: $area_{circle} = \pi r^2$
or
$circumference_{circle} = 2 \pi r$
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 $f(x) = e^x$ is equal to the y-value of the point.