Transcript Base e and Natural Logarithms 10.5

Base e and
Natural Logarithms
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).
Calculating
The value of (1 + 1/n)n approaches e as n
gets bigger and bigger:
(1 + 1/n)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
 1
1  
 n
n
• 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
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
Ex 1
e
0.5
Use a calculator to estimate
to four decimal places.
Ex 2
e
8
Ex 3
Ex 4
ln 3
1
ln
4
Writing Equivalent Expressions
Exponential logarithmic
Write an equivalent equation in the other form.
Ex 5
Ex 6
e x  23
e
x
Ex 7
Ex 8
ln x  1.2528
ln x  2.25
Inverse Properties
e
ln x
x
ln e  x
x
Writing Equivalent Expressions
Ex 9
Evaluate
Ex 10
e
ln 21
Ex 11
Evaluate
ln e
x2 1
Evaluate
e
ln  x 3
Ex 12
Evaluate
ln e
7
Solving Equations
Ex 13
2 x
Solve the following equations.
3e  4  10
Ex 14
2 x
2e  5  15
Solving Equations
Ex 15
Solve the following equations.
ln 3 x  0.5
Ex 16
ln  x  3  3
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!
Examination-style question
The function f is defined by
f ( x ) = 3e x  1  4
x
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.
Examination-style question
d) y = f –1(x) is a reflection of y = f (x) in the line y = x.
x = –4
y y = f(x)
y=x
(0, 4.15)
(–0.71, 0)
–1
(4.15, 0) y = f (x)
x
(0, –0.71)
y = –4