$\log_a ab^2+\log_b a^2b$

Algebra Level 2

If the two roots of the quadratic equation $x^2-6x+1=0$ are $\log a$ and $\log b,$ what is the value of $\log_a ab^2+\log_b a^2b?$

100 90 80 70

2 solutions

Anshuman Harshwardhan
Mar 5, 2014

Roots work out to 3+2sqrt(2) and 3-2sqrt(2)

Expression can be simplified to 2+2(logb/loga + log a/log b)

Mayank Holmes
May 16, 2014

put log a = x and log b = (1/x) ................ ( since log a * log b = 1 ). now, since log a + log b = 6 .... x + (1/x) = 6 , this implies X= 3 (+ 0r - ) 2((2)^(1/2)). if you deduce further you will notice that
log a and log b are equal to one of the one of the solutions............. some calculating manipulation is required further and you just get the answer

log ⁡ a a b 2 + log ⁡ b a 2 b \log_a ab^2+\log_b a^2b lo g a ​ a b 2 + lo g b ​ a 2 b

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$\log_a ab^2+\log_b a^2b$