Solution To Exercises in Logarithm

The following exercises have been extracted from the New General Mathematics for Senior Secondary Schools Book 3. 

Solve the Following equations for x.
(a) log10x=3

We simply use the basic rule of logarithm to solve the question which is P=logbNN=bP

=103=x
x=1000

(b)  logx27=3

x3=27

x3=33

x=3

(c)  3x=1
 Remember from the laws of indices a0=1
3x=30
x=0

(d)  4x+1=8x

we simply equate the bases, so that we can cancel out the bases

22(x+1)=23x
cancel out the bases
2x+2=3x
collect like terms and make x subject of formula
x=2

(e)  2x×23x1=1
by the law addition law of indices, this will become
2x+(3x1)=1

remeber a0=1
24x1=20
cancel out the bases
4x1=0
4x=1
x=14

(f)   22x3×2x+2=0
We collect like terms and use law of indices to simplify bases
22x+x3+2=0
23x1=0
23x=1
23x=20
3x=0
x=0

(g)  6log(x+4)=log64
Using the power law of Logarithm
log(x+4)6=log26
(x+4)6=26
x+4=2
x=2

(h)  92x+1=81x23x
We simply equate the bases of the L.H.S and R.H.S
32(2x+1)=34(x2)3x
34x+2=33x8
4x+2=3x8
x=10

(i)  log10(2x+1)log10(3x2)=1
Remember log101=1
log10(2x+1)log10(3x2)=log10
log10(2x+13x2)=log10
2x+13x2=10
Cross multiply
2x+1=10(3x2)
2x+1=30x20
2x30x+20+1=0
28x+21=0
28x=21
x=2128=34

(j)  3×91+x=27x
Simplify the bases in such a way that they are all the same
3×32(1+x)=33x
31+2+2x=33x
Cancel out the bases
1+2+2x=3x
Collect like terms
3=3x2x
3=5x
x=35

(k)
  log10(3x1)log102=3
Remember log103=log1000=3
log10(3x12)=log103
3x12=1000
Cross Multiply
3x1=2000
Collect like terms
3x=1999
x=19993
x=666.3