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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