Solving Equations Using Logarithms

Edexcel IAL WMA12/01/P2/June/Oct 2020/Q9 (Logarithm & Exponentials)

• Find the exact value of x for which​​ 

log3⁡x + 5–4 =log3⁡2x – 1 

(4)​​ 

• Given that​​ 

3y+3×21– 2y= 108 

• show that​​ 

0.75y = 2 

(4)​​ 

• Hence find the value of y, giving your answer to 3 decimal places.​​ 

(2)

SOLUTION

i- Usng the laws of logarithm.

log3⁡x+5-log3⁡2x-1=4

Applying the division rule.​​ 

log3⁡ x+52x-1 =4 

34=x+52x-1

81=x+52x-1

162x-81=x+5

161x=86

x=86161

x=86161

ii- a-​​ 

3y+3×21– 2y=108

34 x 33 x222y=108

54 x 344y=108

344y=10854

34y=2

0.75y=2

ii- b-​​ 

0.75y=2

Writing this exponential equation into log equation. Thereofre, the base in of the indices becomes the base of log. (Remember,​​ loga⁡b=k​​ can be converted into exponential expression as​​ b=ak).

y=log0.75⁡2

y= -2.4094

y= -2.409

log0.75⁡0.75y=log10⁡2

y=log10⁡2log10⁡0.75

y=-2.409