Question 57
Find the number of real number solutions to the following equation $$ \log _9 {(Y+6)^2} + \log _{27} {(11-Y)^3} = 4$$
Solution
$$ \log _9 {(Y+6)^2} + \log _{27} {(11-Y)^3} = \log _3 {(Y+6)} + \log _3 {(11-Y)} = \log _3 {(66+5Y-Y^2)}$$
So, $$ 66+5Y-Y^2 = 3^4 = 81$$
Or, $$Y^2-5Y+15=0$$
This equation has no real roots
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