We have to start by simplifying using the rule alogn = logn^aalogn=logna.
logsqrt(x - 8) + logsqrt(2x + 1) = 1log√x−8+log√2x+1=1
Now, use the rule logn + logm = log(n xx m)logn+logm=log(n×m)
log(sqrt(x - 8) xx sqrt(2x + 1)) = 1log(√x−8×√2x+1)=1
log(sqrt(2x^2 - 15x - 8)) = 1log(√2x2−15x−8)=1
sqrt(2x^2 - 15x - 8) = 10^1√2x2−15x−8=101
(sqrt(2x^2 - 15x - 8))^2 = (10^1)^2(√2x2−15x−8)2=(101)2
2x^2 - 15x - 8 = 1002x2−15x−8=100
2x^2 - 15x - 108 = 02x2−15x−108=0
2x^2 - 24x + 9x - 108 = 02x2−24x+9x−108=0
2x(x - 12) + 9(x - 12) = 02x(x−12)+9(x−12)=0
(2x+ 9)(x - 12) = 0(2x+9)(x−12)=0
x = -9/2 and 12x=−92and12
However, x = -9/2x=−92 is extraneous since it renders the square root negative, which is undefined in the real number system. Hence, the solution set is {12}{12}.
Hopefully this helps!
