Solving Equations Using Logarithms

Edexcel IAL WMA12/01/P2/Jan 2023/Q4 (Logarithmic​​ Equations,​​ Series Sigma Notation)

• Using the laws of logarithms, solve

log3⁡4x + 2 =log3⁡5x + 7

(3)

• Given that

∑r=12loga⁡yr=∑r=12loga⁡ yr       y > 1, a > 1, y ≠ a 

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

(3)

SOLUTION​​ 

i- Using the laws of logarithms to find the value of x.​​ 

2=log35x+7-log34x

(Applying the division rule.)

2=log35x+74x

32=5x+74x

9=5x+74x

36x=5x+7

31x=7

x=731

ii- ​​ Solving both sides of the equation with the help of sigma notation rule.​​ 

∑r=12logayr= ∑r=12loga yr

(Solving both sides of the equation​​ separately.)

∑r=12logayr=  loga y+logay2

∑r=12loga yr=  loga⁡ y1+loga⁡y2

loga⁡y+loga⁡y2=loga⁡y+loga⁡y2

(Apply the power rule and write 2 as the coefficient of​​ loga⁡y.)​​ 

loga⁡y+2loga⁡y=loga⁡y+loga⁡y2

3loga⁡y=loga⁡y+loga⁡y2

3loga⁡y-loga⁡y=loga⁡y2

2loga⁡y=loga⁡y2

(We can’t apply the power rule on​​ loga⁡y2because the square is not the power of​​ y​​ instead it is the power whole term​​ loga⁡y.) ​​ 

2=loga⁡y2loga⁡y

2=loga⁡y

a2=y

Hence,​​ 

y=a2