Solve the following logarithmic equations and inequalities: a. log5(x+1)−log5(x−1)=2 b. 12log⁡(x4)−log⁡(2x−1)=log⁡(x2)+log⁡2 c. log123+log12x≤log125+log12(x−2)

Question

Solve the following logarithmic equations and inequalities:  a. log5(x+1)−log5(x−1)=2  b. 12log⁡(x4)−log⁡(2x−1)=log⁡(x2)+log⁡2  c. log123+log12x≤log125+log12(x−2)

Theodore Schwartz · Accepted Answer

a. Given: log5(x+1)−log5(x−1)=2 Apply logarithmic property: logma−logmb=logmab log5(x+1x−1)=2 Again, apply logarithmic property: logma=c⇒a=mc, x+1x−1=52 x+1=25(x−1) 26=24x x=1312 b. Given: 12log⁡(x4)−log⁡(2x−1)=log⁡(x2)+log⁡2 Apply logarithmic property: xlogma=logmax, log⁡(x4)12−log⁡(2x−1)=log⁡(x2)+log⁡2 log⁡(x2)−log⁡(2x−1)=log⁡(x2)+log⁡2 Again, apply logarithmic property: logma−logmb=logmab and logma+logmb=logmab, log⁡(x22x−1)=log⁡(2x2) Remove log to both sides of the equation: x22x−1=2x2 x2=4x3−2x2 4x3−3x2=0 x2(4x−3)=0 x=0,34 But for log⁡(x4) and

Solve the following logarithmic equations and inequalities: a. \log_{5}

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