Question

Find the value of x if the base is 10 :

5^logx – 3^{log x –1} = 3^{log x + 1} – 5^{log x –1}

(a) 1 

(b) 0 

(c) 100 

(d) 10

Answer 1

(c) 100

5^{log x} – 3^{log x –1} = 3^{log x + 1} – 5^{log x –1} 

⇒ 5^{log x} – 3^{log x} x 3^–1 = 3^{log x} . 3 – 5^{log x} . 5^–1

⇒ 5^{log x} – \(\frac{1}{3}\)3^{log x} = 3 x 3^{log x} - \(\frac{1}{5}\) x 5^{log x}

⇒ \(\bigg(3+\frac{1}{3}\bigg)\)3^{log x} = \(\bigg(5+\frac{1}{5}\bigg)\)5^{log x} ⇒ \(\frac{10}{3}\) x 3^{log x =} \(\frac{6}{5}\)x 5^{log x}

⇒ \(\frac{3^{log\,x}}{5^{log\,x}}\) = \(\frac{6}{5}\) x \(\frac{3}{10}\) = \(\frac{9}{25}\) ⇒ \(\big(\frac{3}{5}\big)^{log\,x}\) = \(\big(\frac{3}{5}\big)^2\)

⇒ log₁₀ x = 2 ⇒ x = 10² = 100