are in arithmetic progression. Find the value of 
.
and 
such that![\[
\frac{11}{\log_2 x} + \frac{1}{2 \log_{25} x} - \frac{3}{\log_8 x} = \frac{1}{\log_b x} \quad \text{for all } x > 1.
\]](http://latex.artofproblemsolving.com/2/d/7/2d76319b3d4fcd1f193ea0f9f37cb4bf513376f3.png)
![\[ x^{\log{x}} = 10^{2 - 3\log{x} + 2(\log{x})^2}\]](http://latex.artofproblemsolving.com/b/f/b/bfb8185992867a8926b5c62198a799c935771d53.png)
where 
is the logarithm of 
to the base 
.
Prove that
that satisfies![\[
a^3+b^3=1957
\]](http://latex.artofproblemsolving.com/5/d/8/5d8d2a40734ec6b0a9149f9a4ae3a5828d5055f3.png)
![\[
(a+b)(a+1)(b+1)=2014
\]](http://latex.artofproblemsolving.com/3/0/0/300f2d16e669a1d8d9c54b357a6eaf7a8eb5d826.png)
Hence, find the value of 
such that 
is the cube of a positive integer. Find 
with respect to 
is greater than 0 if 
and 
.
and since 
?
Y by
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