Solving Equations Using Logarithms

Edexcel IAL WMA12/01/P2/Oct 2022/Q10 (Logarithms & Exponential Equations)

Given​​ a =log2⁡3 

• write, in simplest form, in terms of a,​​ 

• log2⁡9​​ 

• log2⁡316​​ 

(3)

• Solve​​ 

3x ×2x+4 = 6 

giving your answer, in simplest form, in terms of a.​​ 

(4)

SOLUTION

I- Use the laws of logarithm, to give the answer in terms of a.​​ 

a-​​ 

log2⁡9 =log2⁡32

log2⁡9=2log2⁡3

Since​​ a=log2⁡3

log2⁡9=2a

b-​​ 

log2⁡316=log2⁡3-log2⁡ 16

=log2⁡312 -log2⁡24 

=12log2⁡3-4log2⁡2

log2⁡316=12 a-4

ii- Applying the laws of logarithm, to solve the equation.​​ 

log2⁡3x x 2x+4=log2⁡6

Applying the mulplication rule here.​​ 

log2⁡3x+log2⁡2x+9 =log2⁡6

We will write the argument of​​ log2⁡6​​ into (2×3), so that we may get the answer in terms of a Since​​ a=log2⁡3.​​ 

log2⁡3x+log2⁡2x+9 =log2⁡2 x 3

xlog2⁡3+x+4log2⁡ 2=log2⁡2+log2⁡3

Putting​​ a=log2⁡3

xa+x+4=1+a

ax+x=1+a-4

x a+1=a-3

x=a-3a+1