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Logarithms

Express each of the following in the form in x=ax + b, and find the value of a and of b.
 
(i) xe-x =2.46
(ii) (xex)2 = 30e-x
 
 
 
Ans:(i) a=1, b= 0.9
       (ii) a= -3/2, b=1.70
 
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1 Answer

"Express each of the following in the form ln(x)=ax + b, and find the value of a and of b."
 
(i) xe^(-x) = 2.46
ln(x) - x(ln(e)) = ln(2.46)
ln(x) - x = ln(2.46)
ln(x) = x + ln(2.46)
a = 1, b = ln(2.46)

(ii) (xe^x)^2 = 30e^(-x)
(x^2)(e^(2x)) = 30e^(-x)
ln(x^2)+ln(e^(2x)) = ln(30) + ln(e^(-x))
2 ln(x) + 2x ln(e) = ln(30) - x ln(e)
2 ln(x) + 2x = ln(30) - x
2 ln(x) = -2x + ln(30) - x
2 ln(x) = -3x + ln(30)
ln(x) = -3x/2 + (ln(30))/2
a = -3/2, b = (ln(30))/2