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Answer 1 · 2015-06-21T19:00:49

The solution is : $x = e \lor x = \frac{1}{e}$ $x=e vv x=1/e$

The logarythms have different bases, so first you have to change base using formula:

${log}_{a} x = \frac{{log}_{b} a}{{log}_{b} x}$ $log_ax=log_ba/log_bx$

In this case we get:

${log}_{e} x - \frac{1}{{log}_{e} x} = 0$ $log_ex-1/log_ex=0$

${({log}_{e} x)}^{2} - 1 = 0$ $(log_ex)^2-1=0$

$({log}_{e} x - 1) ({log}_{e} x + 1) = 0$ $(log_ex-1)(log_ex+1)=0$

${log}_{e} x - 1 = 0 \lor {log}_{e} x + 1 = 0$ $log_ex-1=0 vv log_ex+1=0$

${log}_{e} x = 1 \lor {log}_{e} x = - 1$ $log_ex=1 vv log_ex=-1$

$x = e \lor x = \frac{1}{e}$ $x=e vv x=1/e$