I am stuck on two equations and can not seem to get this undermy belt 1. log x +

Question

I am stuck on two equations and can not seem to get this undermy belt 1. log x + log (x + 3) = 1 2. log3 (2x – 1) – log3 (x – 4) = 2 These are the notes I have: Read "log 5 x" as "log to the base 5 of the value x ", "log3"as "log to the base 3" and "log" as "log to the base 10". I am stuck on two equations and can not seem to get this undermy belt 1. log x + log (x + 3) = 1 2. log3 (2x – 1) – log3 (x – 4) = 2 These are the notes I have: Read "log 5 x" as "log to the base 5 of the value x ", "log3"as "log to the base 3" and "log" as "log to the base 10".

Explanation / Answer

1. log x + log (x + 3) = 1 for the log to exist we must have x>0 and x+3>0                                                              =>x>-3 Hence we have logx(x+3)=1 using (logab=logga+logb)                     =>x(x+3)=10                     =>x2+3x-10=0                     =>x2+5x-2x-10=0                     =>x(x+5)-2(x+5)=0                     =>(x+5)(x-2)=0                    =>x=-5 or x=2           butx>0=>x=2 2. log3 (2x – 1) – log3 (x – 4) = 2 we must have 2x-1>0 and x>4                   =>x.0.5 and x>4 Now log3(2x-1)(x-4)=2         =>(2x-1)(x-4)=32=9         =>2x2-9x-5=0         =>2x2-10x+x-5=0         =>2x(x-5)+(x-5)=0        => (x-5)(2x+1)=0        =>x=5 or x=-0.5 but x>4=>x=5

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