To solve the given equation, w need to review some rules:
[tex](1) \ ln \ e = 1 \\
(2) \ ln \ b^{a} = a*ln \ b \\
(3) \ ln \ a + ln \ b = ln \ (ab) \\
(4) \ ln \ a - ln \ b = ln \ \frac{a}{b} \\[/tex]
The given equation is :
[tex]ln \ e^{ln \ x} + ln \ e^{ln \ x^{2}} = 2 \ ln \ 8[/tex]
[tex]ln \ x * ln \ e + ln \ x^{2} * ln \ e = 2 \ ln \ 8 [/tex] ⇒⇒⇒⇒ rule (2)
[tex]ln \ x + ln \ x^{2} = ln \ 8^{2} [/tex] ⇒⇒⇒⇒ rule (1) and rule (2)
[tex]ln \ ( x* x^{2} ) = ln \ 64[/tex] ⇒⇒⇒⇒ rule (3)
removing the nature logarithm from both sides
[tex] x^{3} = 64 = 4^{3} [/tex]
∴ x = 4
So, the correct answer is option (2) ⇒⇒⇒⇒ x = 4
