An algebra problem by Florin Celoiu

Algebra Level 3

$\Large n^{5^{ (\log_{5}2) + (\log_{5}x )+ \log_{5}\left(1-\frac{9}{2x}\right) }}=n^{7^{(\log_{7}56) - \log_{7}x}}$

For $n \geqslant 2$ , what is the positive integral value of $x$ which satisfies the above equation?

1 solution

Florin Celoiu
Dec 7, 2015

The first part of equation can be rewritten like: $n^{5^{log_{5}2 + log_{5}x + log_{5}(1-\frac{9}{2x})}} = n^{2x-9}$

The second part of equation can be rewritten like: $n^{7^{log_{7}56 - log_{7}x}} = n^{\frac{56}{x}}$

Now we have the following equation: $n^{2x-9} =n^{\frac{56}{x}}$

Then we apply $log_{n}$ in both sides and the equation will become: $2x-9 = \frac{56}{x}$

$2x^{2} - 9x - 56 = 0$

If we resolve this quadratic equation we will obtain two solutions, but only one it is a positive integer number (8).