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Exponential Equation

Question 4

Solving the equation x^{lg x} = 1000 x²

Solution:

From the definition of logarithm, if a^x = b (a > 0, and a is not 0) then x = log_ab

If a = 10, then log₁₀b, marked as lg b,

Take the lg of both side ( note: lg b is log₁₀b)

lg(x^{lg 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:

log_ab^x = x log_ab

log_aa = 1

log_a(MN) = log_aM + log_aN