How this form can be induced? (BJT accurate value of vbe)
You can derive this from the Shockley equation:
*Vbe  = Vt(ln(Ic/Is)) so at two different currents Ic and Ic0 we have
\$V_{BE} - V_{BE0} \$ = \$V_T (\ln(I_C/I_S) - \ln(I_{C0}/I_S))\$ = \$V_T \cdot \ln (I_C/I_{C0})\$
At room temperature Vt = kT/q is about 25mV.
By using two currents, the saturation current Is cancels out, so you can predict the behavior independent of the device parameters.
* We can ignore the factor of 1 in the Shockley equation at reasonable currents &gt;&gt; Is.