Base e and Natural
Logarithms
HistoryThe number e is a famous irrational number, and is one of the most important numbers in mathematics. The first few digits are2.7182818284590452353602874713
527...
It is often called Euler's number after Leonhard Euler. e is the base of the natural logarithms (invented by John Napier).
CalculatingThe 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
Vocabularynatural base: the number e, which is found using
• the base rate of growth shared by all continually growing processes
• Used heavily in science to model quantities that grow & decay continuously
natural base exponential function: an exponential function with base e
11
n
n
Vocabularynatural 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
Ex 1
Use a calculator to estimate to four decimal places.
0.5e 8eEx 2
Ex 3 Ex 4ln 3 1
ln4
Ex 5
Exponential logarithmicWrite an equivalent equation in the other
form.
23xe
Ex 6
Writing Equivalent Expressions
xe
Ex 8Ex 7
ln 1.2528x ln 2.25x
Inverse Properties
ln xe x ln xe x
Ex 9
Evaluate2 1ln xe
Ex 11
Writing Equivalent Expressions
Evaluate
ln 21e
Ex 10Evaluat
e
ln 3xe
Evaluate 7ln eEx 12
Solving Equations
23 4 10xe Ex 13
Solve the following equations.
22 5 15xe Ex 14
Solving Equations
Ex 15Solve the following equations.
Ex 16
ln 3 0.5x ln 3 3x
Graphing properties( and lnx are inverse
functions reflected in y=x)
These can be transformed (stretched, translated and reflected)in the same way as other functions. But, be careful of the hidden asymptotes!
xe
Examination-style questionThe function f is defined by
a) Describe the sequence of geometrical transformations by which the graph of y = 3ex + 1 – 4 can be obtained from that of y = ex.
b) The graph of y = f(x) crosses the y-axis at point A and the x-axis at point B. Write down the coordinates of A and B, working to 2 decimal places.
c) Write an expression for f –1(x) and state its domain and range.d) Sketch the graphs of y = f(x) and y = f –1(x) on the same set of axes
and state their geometrical relationship.
f 1( ) = 3 4xx e x
Examination-style questiond) y = f –1(x) is a reflection of y = f (x) in the line y = x.
y = x
x
y
y = –4
y = f(x)
y = f –1(x)
x = –4
(–0.71, 0)
(0, –0.71)
(0, 4.15)
(4.15, 0)
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