Question

Solve each equation for x. (a) 5^(x−5)=2 (b) lnx+ln(x−1)=1?

Answer 1

(a) 5^(x−5)=2 then log_5 (5^(x−5) )=log_5 (2)
x-5=log_5 (2) x=5+log_5
(2)

(b) ln(x)+ln(x-1)=1 we have that x>1 from definition of
log
ln(x)+ln(x-1)=ln( x*(x-1) )=1
e^ln( x*(x-1) )
=e^1
x^2-x-e=0
D=1+4e
x1=(1+sqrt(1+4e) )/2
x2=(1-sqrt(1+4e)
)/2

x2<1 so it cant be a root so x1=(1+sqrt(1+4e) )/2 is a solution