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Question

4(ln(x))^(2) − ln(x^(7)) − 15 = 0

Answer 1

The logarithmic equation is 4(ln x)² - ln (x⁷) - 15 = 0.

Apply power property of logarothm : ln (xⁿ) = n ln x.

4(ln x)² - 7ln (x) - 15 = 0

Let, ln x = u,

Then, the equation is 4u² - 7u - 15 = 0

4u² - 12u + 5u - 15 = 0

4u(u - 3) + 5(u - 3) = 0

(u - 3)(4u + 5) = 0

u  - 3 = 0   and     4u + 5 = 0

⇒ u = 3 and u = - 5/4

put, u = ln x.

⇒ ln x = 3  and  ln x = - 5/4

Apply exponential property of logarithm : ln x = n ⇔ x = eⁿ .

⇒ x = e³  and  x = e^{(- 5/4)}

x = 20.0855 and  x = 0.2865.

The solution is x = 0.2865 , x = 20.0855 .