Question

If \(x\) and \(y\) are positive real numbers such that \(log_x(x^2+12)=4\) and \(3\;log_yx=1\),then \(x+y\) equals

• A. 11
• B. 20
• C. 10
• D. 68

Answer 1

The Correct Option is C

We have \(\log_x(x^2 + 12) = 4\)
\(⇒  x ^2 +12=x ^4 \)
\(⇒x^ 4 −x^ 2 −12=0 \)
\(x^2(x^2 - 4) + 3(x^2 - 4) = 0\)
 \((x ^2 −4)(x^ 2 +3)=0\)
given that x is a positive real number, then x = 2.

Given   \(3\log_y{x} = 1\)
\(\log_y{x} = \frac{1}{3}\)
\(⇒\)   \(x = y^\frac{1}{3}\)
\(⇒\)   \(y=x^ 3 \)
\(⇒\) \(y = 8.\)
\(⇒\) \(x + y = 2 + 8 = 10.\)