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Exponential Equation

Question 4
Solving the equation xlg x = 1000 x2
Solution:
From the definition of logarithm, if ax = b (a > 0, and a is not 0) then x = logab
If a = 10, then log10b, marked as lg b,
Take the lg of both side ( note: lg b is log10b)
lg(xlg x) = lg(1000 x2)
(lg x)(lg x) = lg1000 + lg x2
(lg x)2 = lg 103 + 2 lg x
(lg x)2 - 2 lg x - 3 = 0
Factoring this quadratic equation, (lg x + 1)(lg x - 3) = 0
lg x + 1 = 0 or lg x - 3 = 0
For lg x + 1 = 0 => lg x = -1 => 10-1 = x => x = 1/10
For lg x - 3 = 0 => lg x = 3 => 103 = x => x = 1000
formulas used:
logabx = x logab
logaa = 1
loga(MN) = logaM + logaN