F.4 MATHS (logarithmic)

1)
3²ˣ⁺¹ + 3²ˣ⁻¹ = 8
3²(3²ˣ⁻¹) + 3²ˣ⁻¹ = 8
3²ˣ⁻¹(3² + 1) = 8
3²ˣ⁻¹(10) = 8
log[3²ˣ⁻¹(10)] = log8
log3²ˣ⁻¹ + log10 = log8
(2x - 1)log3 = -1 + log8
2x - 1 = (-1 + log8)/log3
2x = 1 + [(-1 + log8)/log3]
x = (1/2) + [(-1 + log8)/2log3]
x = 0.398


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2)
logy = log(2x) - 1 …… (1)
logx = 2log√(y + 1) …… (2)

(1):
logy = log2x - log10
logy = log(2x/10)
y = 2x/10
x = 5y ….. (3)

(2):
logx = 2log(y + 1)¹ʹ²
logx = (2)(1/2)log(y + 1)
logx = log(y + 1)
x = y + 1 …… (4)

(3) = (4):
5y = y + 1
4y = 1
y = 0.25

Put y = 0.25 into (4):
x = 0.25 + 1
x = 1.25

Hence, x = 1.25, y = 0.25


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3)
3log92 - log954 - log936
= log92³ - log954 - log936
= log98 - log954 - log936
= log8/log9 - log54/log9 - log36/log9
= (log8 - log54 - log36)/log9
= log[8/(54*36)]/log9
= log(1/243)/log9
= -log243/log9
= -log(3)⁵/log(3)²
= -5log3/2log3
= -5/2
= -2.5


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1)
log3(3ˣ⁺¹ - 18) = x
3ˣ⁺¹ - 18 = 3ˣ
3ˣ⁺¹ - 3ˣ = 18
(3)3ˣ - 3ˣ = 18
3ˣ(3 - 1) = 18
3ˣ(2) = 18
3ˣ = 9
3ˣ = 3²
x = 2

2. From the first equation,
logy = log2x - 1
logy = log2x - log10
logy = log(x/5)
y = x/5---------------------(1)
From the second equation,
logx = 2logroot(y + 1)
logx = log(y + 1)
x = y + 1------------------(2)
Soving (1) and (2), x = 5/4, y = 1/4

F.4 MATHS (logarithmic)Find the unknown

1) 3^2x+1+3^2x-1=8(cor. to 3 s.f.)


2)
圖片參考：http://imgcld.yimg.com/8/n/HA00749348/o/701011160144213873411420.jpg


3) 3 log9 2 - log9 54 –log9 36


Simplify the equation
1) log3 (3^(x+1) -18)=x Ans:
1) 3^2x+1+3^2x-1=8
(2x+1)log3+(2x-1)log3=log8
4xlog3=log8
x=log8/4log3
=0.473 (cor. to. 3 s.f.)

2)
logy=log(2x)-1
log(y/2x)=-1
y/2x=0.1
y=0.2x .....(1)
2logx=2log(y+1)^0.5

log x^2=log(y+1)
x^2=y+1
y=(x^2)-1 ....(2)
(1)=(2):y=0.2x=(x^2)-1
(x^2)-0.2x-1=0
5(x^2)-x-5=0 ....(3)
x= -0.905(rej.) or 1.10 (cor .to 3 s.f.)
Put x =1.10 into (1),
y=0.220 (cor. to 3 s. f.)
(x,y)=(1.10,0.220)

3)3 log9 2 - log9 54 –log9 36

=log9 8-log9 54-log 936
=log9(8/(54*36))
=log9 (1/243)
=-5log3/2log3
=-2.5

log3 (3^(x+1) -18)=x

3^(x+1) -18=3^x
3^(x+1) -3^x-18=0
2(3^x)=18
3^x=9
x=2

2. y=log(2x)-1+log
log x=2log log(2x)-1+log
logx =3log(2x)-1+log
logx=6log
x=6