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Lebl-Basic-Analysis-I
· Problem 5.4.10
The Riemann Integral
Mathematics · logarithm-exponential
Problem
Show that
E
(
x
)
=
e
x
E(x) = e^x
E
(
x
)
=
e
x
is the unique continuous function such that
E
(
x
+
y
)
=
E
(
x
)
E
(
y
)
E(x+y) = E(x)E(y)
E
(
x
+
y
)
=
E
(
x
)
E
(
y
)
and
E
(
1
)
=
e
E(1) = e
E
(
1
)
=
e
. Similarly, prove that
L
(
x
)
=
ln
(
x
)
L(x) = \ln(x)
L
(
x
)
=
ln
(
x
)
is the unique continuous function defined on positive
x
x
x
such that
L
(
x
y
)
=
L
(
x
)
+
L
(
y
)
L(xy) = L(x) + L(y)
L
(
x
y
)
=
L
(
x
)
+
L
(
y
)
and
L
(
e
)
=
1
L(e) = 1
L
(
e
)
=
1
.
Related theory
The Complex Exponential And Logarithm
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