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| Transcendental
functions - The
graphs of transcendental functions |
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· Exponential
and logarithmic functions are mutually inverse functions |
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- Exponential
function |
y
= ex
<=>
x = ln y,
e = 2.718281828...the
base of the natural logarithm, |
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exponential
function is inverse
of the natural logarithm
function, so that eln
x = x. |
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- Logarithmic
function |
y
= ln x
= loge x
<=>
x = e y,
where x
> 0
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the
natural logarithm
function is inverse
of the exponential
function, so that
ln(ex) =
x. |
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- Exponential
function |
y =
ax
<=>
x = loga
y,
where a > 0 and
a
is
not 1 |
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| exponential
function with base a is
inverse of the logarithmic
function, so that |
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- Logarithmic
function |
y =
loga
x
<=>
x = a y,
where a
> 0,
a
is
not 1
and x
> 0 |
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the
logarithmic
function with base a
is inverse of the exponential
function, so that
loga(ax) =
x. |
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| The
graph of the exponential
function y
= ax = ebx,
a
> 0
and b
= ln a |
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| The
exponential
function is
inverse of the logarithmic
function since its
domain and the range are respectively the range and domain of
the
logarithmic
function and |
| ƒ(f
-1(x))
= x that is, ƒ(f
-1(x))
= ƒ(ax) = loga(ax)
=
x. |
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| The
graph of the logarithmic
function y
= logax,
a
> 0
and for a
= e,
y
= logex = ln x |
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| The
logarithmic
function is
inverse of the exponential
function since its
domain and the range are respectively the range and domain of
the
exponential function and |
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