C2Logarithms - Mathshelper

1 C2 LogarithmsPatrons are reminded that if you see a log with no base then it means log10. For example log 7 log107. You are also reminded that if you have a number which is not alog (3, say) and you wantto write it in terms of a logarithm then this is how to do it:3=3 1=3 logaa=loga(a3).1. Write down (without a calculator) the value of the following logarithms:(a) (b) (c) (d) (e) log10(1100). 2(f ) log2(116). 4(g) loga(a6).6(h) log a(a2).42. State (without a calculator) twoconsecutiveintegers that the following logarithms lie between:(a) (b) (c) (d) 1and03. Express the following as a single logarithm:(a) logax+loga(x2).logax3(b) log2(x3) log2(x2).log2x(c) logca+logc(ab).logca2b(d) 2 logx+3 (e) 2 log5x log5y+ (x2zy)(f ) 2 log7(s2) 3 log7(s3)+5 (t5s5)(g) loga 3 logb 7 logc+ (10ab3c7)(h) 2 logap+logaq 7 logar (p2qa3r7)4. Solve the following equations (if there is a logarithm in brackets after the question, please uselogarithms tothatbase to solve the problem, even if it is unnatural to use that base).

2 Give allanswers to three significant figures, where appropriate.(a) 2x=5.(log10)x= (b) 8x=3.(log8)x= (c) 32x=11.(log3)x= (d) 53x 4=100.(log10)x= (e) 17=13x 4.(log5)x= (f ) 2x2x+1=10.(log2)x= (g) 5=7 2x+1.(log10)x= (h) 3 22 3x=13.(log3)x= (i)a bx+c=d.(logd)x=1 logda clogdblogdb(j) 11 9x=13.(log3)x= (k) 3x=2x+1.(log5)x= (l) 3x+1=42x 1.(log4)x= (m) 2 3x=51 x.(log10)x= (n) 7 22x+1=6 11x+1.(log2)x= (o)a bcx+d=e fgx+k.(logz)x=logze logza+klogzf dlogzbclogzb Solve the following equations (you may need the factor theorem for the later problems):(a) log2x log2(x 1)= (b) log3(x+2)+log3x= (only)(c) log3(2x) log3(1 x)= (d) 2=log2(2x)+log2(x 1).x=2(only)(e) log2x+log2(2x+1)= 513 14(only)(f ) log2(x 1)=4+log2(2x+3).No solns(g) 2 log5x+log5x= (h) 2 log2(x+3)+log2x log2(4x+2)= 41 52(only)6. Solve the following equations:(a) 22x+15=8 (b) 8 3x=32x+ (c) 52x=16 6 (only)(d) 4 7x+72x+3= solns(e) 3x+1= (only)(f ) 3 22x+5=16 (13)orx=log25(g) 4 32x=35+4 (72)(only)(h) 22x+35=3 2x+ Given thatx=logapandy=logaq, write the following in terms ofxandy(a) loga(p2q).

3 2x+y(b) loga(q p).y x2(c) loga(p2q) 2 loga(qp).4x y(d) loga( pq3) 12loga(qp).y(e) (f ) Find the intersection of the curvesy=log2x+3 andy=log2(x+3).(x,y)=(37,log224 log27)9. (a) Show that if loga+logc=2 logbthena,bandcare in geometric progression.(b) Show that if logx+logz=3 logythenx,y2andyzare in geometric The definition of a logarithm is given bya=bc c=logba.(a) Takea=bcand this time take logs to the basecof both sides of the equation and henceprove thatlogcb logba=logca.(b) Hence or otherwise calculate to 4 significant figures log35.(c) Deduce log325 and log3( 53).11. Taking the same scale on thexandy-axes, draw a separate sketch for each of the following:(a)y=log2x.(b)y=log2( x).(c)y=log2(x+3).State howy=log2xcan be transformed into each of the other (a) Write each of 169 and 243 as a product of prime numbers.(b) Writex=log3169 in index form.

4 (c) Evaluate log3169 log13243 without using a A firm is testing two types of scrubbing brush by using a machine that keeps the brushes incontinuous action.(a) The first brush starts with 2000 bristles and the number ofbristles,n, left aftertdays isknown to follow the rulen=2000 2 the number of bristles left after 10 days.(b) The second brush starts with 1450 bristles and follows the rulen=A 3 t/PwhereAandpare constants. After 10 days it is found to have 1373 bristles. Write downthe value ofAand calculate the value ofpto the nearest It is often easy to prove that many logarithms are irrational numbers, and a method of proofmay bereductio ad absurdam(proof by contradiction).For example, consider logmn. Suppose thatmandnare natural numbers ( numbers fromthe set {1,2,3,4,..}) and, first, that one is odd and the other even. Usingreductio ad absurdam,prove that logmnis