Understanding Engineering Mathematics

B.Express in simplest form

(i)

e^2 A−e^2 B

eA+eB

(ii) (eA−e−A)(eA+e−A)

(iii)

e^2 A+ 1

e^2 A+e−^2 A+ 2

(iv) (eA+e−A)^2 −(eA−e−A)^2

4.1.5 Logarithms to general base ➤130 137➤➤

Evaluate (ln denotes logs to basee)

(i) log 101 (ii) log 22 (iii) log 327

(iv) log 2

( 1

4

)

(v) loga(a^4 ) (vi) loga(ax)

(vii) log 31 (viii) lne (ix) ln

√

e

(x) lne^2 (xi) ln 1

4.1.6 Manipulation of logarithms ➤131 137➤➤

A.Express each of the following as a single logarithm (lnxis to basee,logaxto basea)

(i) lnx+2lny (ii) 3 lnx−4lny

(iii) 2 lnx−3ln( 2 x)+4lnx^3 (iv) 3 logax+2logax^2

(v) alogax+3loga(ax)

B. If log 2 x=6, what is log 8 x?
C.If lny=2lnx−^1 +ln(x− 1 )+ln(x+ 1 )obtain an expression foryexplicitly in terms
ofx, stating any conditions required onx,y.

4.1.7 Some applications of logarithms ➤134 138➤➤

A.Solve the equation 2x+^1 =5 givingxto three decimal places.

B. By making an appropriate transformation of the variables convert the equation

y= 3 x^6

to one which has the forms of a straight line – i.e. alinear form. What is the gradient
and the intercept of the line?

4.2 Revision

4.2.1 y=an,n=an integer

➤

119 136➤

The exponential function is essentially a power function in which the exponent is the
variable. As such it obeys all the usual rules of indices. We can get some idea of the