Solving Equations with e and ln x - mit.edu

Solving Equations witheandlnxWe know that the natural log function ln(x) is defined so that if ln(a) =btheneb=a. Thecommon logfunction log(x) has the property that if log(c) =dthen10d=c. It s possible to define a logarithmic function logb(x) for any positivebasebso that logb(e) =fimpliesbf=e. In practice, we rarely see bases otherthan 2, 10 fory:1. ln(y+ 1) + ln(y 1) = 2x+ lnx2. log(y+ 1) =x2+ log(y 1)3. 2 lny= ln(y+ 1) +xSolve forx(hint: putu=ex, solve first foru) +e xex e x= +e xSolutions1. ln(y+ 1) + ln(y 1) = 2x+ equation involves natural logs. We apply the inverseexof the func-tion ln(x) to both sides to undo the natural (y+ 1) + ln(y 1) = 2x+ lnxeln(y+1)+ln(y 1)=e2x+lnxeln(y+1) eln(y 1)=e2x elnx(y+ 1) (y 1) =e2x xy2 1 =xe2xy2=xe2x+ 1y= xe2x+ 1We know that we cannot take the natural log of a negative number (orof 0), and our equation contains the expression ln(y 1).

Solving Equations with e and lnx We know that the natural log function ln(x) is de ned so that if ln(a) = b then eb = a. The common log function log(x) has the property that if …

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