Question

If \(5^{{3x^2}log_{10}2}\)= \(2^{(x+1/2)log_{10}25}\), then the value of x is:

(a) –1 

(b) 2 

(c) \(\frac{1}{2}\)

(d) \(-\frac{1}{3}\)

Answer 1

(d) \(-\frac{1}{3}\)

\(5^{{3x^2}log_{10}2}\) = 2^{\(\big(x+\frac{1}{2}\big)\)log₁₀ 25}

⇒ \(5^{{3x^2}log_{10}2}\) = 2^{\(\big(\frac{2x+1}{2}\big)\) x log₁₀ 5}  = 2^{(2x+1)log₁₀ 5}

⇒ \(5^{{3x^2}log_{10}2}\) = 2^{(2x+1)log₂ 5. log₁₀ 2}           (using log_a x = log_b x . log_a b)

⇒ \(5^{{3x^2}log_{10}2}\) = [\(2^{log_25^{(2x+1)}}\)] ^{log₁₀ 2}

⇒ \(\big(5^{{3x^2}}\big)\)^{log₁₀ 2} = (5^2x+1)^{log₁₀ 2}               [Using a^{log_a x} = x]

⇒ 3x² = 2x + 1 ⇒ 3x² – 2x – 1 = 0 

⇒ (x – 1) (3x + 1) = 0

⇒ x = 1, \(-\frac{1}{3}\)