082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.4 Review of some common engineering functions and techniques 91
Table2.5
Somevalues forlogarithm
functionslogx and lnx.
x logx lnx
y
ln x
log x
0.1 −1 −2.303
0.5 −0.301 −0.693
1 0 0
2 0.301 0.693
5 0.699 1.609
10 1 2.303
50 1.699 3.912
Figure2.32
Graphsoflnx and logx.
1 x
are one-to-one. It is important to stress that the logarithm functions, log a
x, are only
defined for positive values ofx. The following properties should be noted:
}
}
logx→∞ logx→−∞
asx→∞ asx→0
lnx→∞ lnx→−∞
log1=ln1=0
log10=1 lne=1
Connectionbetweenexponentialandlogarithmfunctions
Theexponentialfunction, f (x) =a x ,isaone-to-onefunctionandsoaninversefunction,
f −1 (x), exists.Recall
So
Now
f −1 (f(x))=x
f −1 (a x )=x
log a
(a x ) =xlog a
a
using laws of logarithms
=x since log a
a = 1
Hence the inverse of f (x) = a x is f −1 (x) = log a
x. By similar analysis the inverse of
f(x)=log a
xisf −1 (x) =a x .
The inverse of the exponential function, f (x) = a x , is the logarithm function, that
is f −1 (x) = log a
x.
Theinverseofthelogarithmfunction, f (x) = log a
x,istheexponentialfunction,
thatis f −1 (x) =a x .
In particular:
If f(x) =e x ,then f −1 (x) =lnx.
If f(x) = lnx,then f −1 (x) =e x .
If f(x) = 10 x ,then f −1 (x) =logx.
If f(x) = logx,then f −1 (x) =10 x .