<s>[INST] Simplifying Logarithmic Expression 
Compute:
$$\frac{1-\log_a^{3}{b} }{(\log_a b+\log_b a+1)\log_a\frac{a}{b}}$$

I tried to expand it :
$$\frac{1-\log_a^{3}{b} }{(\log_a b+\log_b a+1)\log_a\frac{a}{b}}$$
$$=\frac{(1-\log_a{b})(\log_a^{2}b+\log_a b+1)}{(\log_a b+\log_b a+1)(1-\log_a{b})}$$
$$=\frac{(\log_a^{2}b+\log_a b+1)}{(\log_a b+\log_b a+1)}$$
But I got nothing.
 [/INST] $$ \large{x =  \log_a b}$$
Then $$ 1- \log_a^{3}{b} = (1-x^3) $$
Also 
$$(\log_a b+\log_b a+1)(\log_a\frac{a}{b}) = (x+\frac{1}{x}+1)(1-x)$$
because
$$ \log_b a = \frac{1}{\log_a b} \hspace{8pt} \textit{and} \hspace{8pt} \log_a \frac{a}{b} = (1-\log_a b)$$
The whole thing gets simplified to
$$ \frac{(1-x^3)}{(x+\frac{1}{x}+1)(1-x)} = \frac{x(1-x^3)}{1-x^3}= x$$
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