How do you solve #(\log _ { 3} ( x ^ { 4} ) ) ^ { 2} - 5\log _ { 3} ( x ^ { 4} ) = - 4#?

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Answer 1 · 2017-12-09T09:50:17

#x={3,3^(1/4) }#

We must first use our knowledge about logarithms;

#beta*logalpha = log alpha^beta #

#=> log_3 x^4 = 4log_3 x #

Hence our equation becomes;

#(4log_3 x )^2 -20log_3 x +4 = 0#

#16(log_3 x)^2 -20log_3 x +4 = 0 #

Now subsitute #log_3 x = mu #

#=> 16mu^2 -20mu +4 = 0 #

Now this is a simple quadratic, using the formula or otherwise yields;

#mu = {1,1/4 } #

Hence:

#log_3 x = {1,1/4}#

#=> x = {3^1 , 3^(1/4)} = {3,3^(1/4)} #