Solving Equations Using Logarithms

Edexcel IAL WMA12/01/P2/Jan 2022/Q4 (Logarithmic​​ Equation)

Using the laws of logarithms, solve​​ 

log3⁡(32 – 12x) = 2log3⁡(1 – x) + 3 

(5)

SOLUTION​​ 

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

Applying the division rule.

log3⁡32-12x-3log3⁡1-x=3

log3⁡32-12x-log3⁡1-x2=3

log3⁡32-12x1-x2=3

32-12x1-x2=33

32-12x=27 1-x2

32-12x=27 1-2x+x2

27x2-42x-5=0 

9x+13x-5=0 

9x+1=0     3x-5=0

x= -19       x=53

Don’t forget to check that which value of x gives the argument of the log equations positive. It​​ is so because the argument of a log function is always positive and should not be equal to 1.

So, to check substituting​​ -19​​ in both argument.​​ 

32 – 12x=32-12-19=1003>0

1 – x=1--19=109>0

So,​​ x= -19​​ accepted.

Now, substituting ​​ x=53​​ in both argument.

32 – 12x=32-1253=12>0

1 – x=1-53=-23<0

So,​​ x= -19​​ rejected.

Hence, the value of x is​​ -19.