Solving Equations with e and ln x

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 eq

input to a logarithmic function; we isolated it by using the exponential inverse of that logarithmic function. In this problem our variable is the input to an exponential function and we isolate it by using the logarithmic function with the same base. ex = r y + 1 y 1 ln(ex) = ln r y + 1 y 1 x = ln " y + 1 y 1 1 2 # 3

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