Solve the equation log4 x 4 log4 x 10 2 log23x 2 log2x

Solve the equation. log_4 (x - 4) + log_4 (x - 10) = 2 log_2(3x - 2) - log_2(x - 5) = 4

log₄(x-4) + log₄(x-10)=2

log₄(x-4) + log₄(x-10)=2 log₄4 (since log₄4=1)

log₄(x-4)(x-10)= log ₄4²

log₄(x-4)(x-10)=log₄16

(x-4)(x-10)=16

x^2-14x+40-16=0

x²-14x+24=0

(x-12)(x-2)=0

x=12,2

ignoring x=2 ,since on using x=2 we get negative log value and log cant take negative values

therefore x=12 is the answer.

2. log₂(3x-2) - log₂(x-5)= 4

log ₂((3x-2)/(x-5))=4 log₂2(since log ₂2=1)

log₂((3x-2)/(x-5))= log ₂2⁴

(3x-2)/(x-5)=16

3x-2=16(x-5)

3x-2=16x-80

-2+80=16x-3x

78=13x

x=6

log₂((3x-2)/(x-5))=log₂16