Understanding Engineering Mathematics

=

eA(eA+e−A)

(eA+e−A)^2

=

eA

eA+e−A

=

e^2 A

e^2 A+ 1

(iv)(eA+e−A)^2 −(eA−e−A)^2 =(eA)^2 + 2 eAe−A

+(e−A)^2 −((eA)^2 − 2 eAe−A+(e−A)^2 )

= 4 eAe−A= 4

4.2.5 Logarithms to general base

➤

120 137➤

What about theinverse function(100
➤
) of the exponential function? That is, if

y=ax

then what isxin terms ofy?
By definition, we call the inverse of the exponential function thelogarithm to basea
and write

x=logay

Note thatamust be positive if we are to avoid complex numbers and it must not be equal
to unity, since 1 raised to any power will again be 1.
In the special case of the exponentialex, we call the inverse thenatural logarithm,
denoted ln. Thus if

y=ex, thenx=logey=lny

An equivalent definition of thelogarithm of a numberxto baseais as that power to
which the base must be raised to givex.Thatis,

x=alogax

In particular

x=elnx

Note that sinceaxcan never be negative, then logayis only defined for positive values of
y. This is sometimes emphasised by writing loga|y|, although often we omit the modulus
signs and simply take it for granted that all real quantities under a logarithm are to be
assumed positive.
From the definition of logaxit also follows that

x=logaax

and in particular

x=lnex