The apparent magnitude [tex]m_i[/tex] of a star i measured with respect to the magnitude [tex]m_{ref}[/tex] of another star is given by
[tex]m_i -m_{ref}=-2.5 \log_{10}( \frac{I_i}{I_{ref}} )[/tex]
where [tex]I_i[/tex] and [tex]I_{ref}[/tex] are the brightnesses of the two stars.
In our problem, the star used as reference has magnitude 5, so [tex]m_{ref}=5[/tex]. We also know that the star i is 10 times brighter than the star of magnitude 5, so
[tex] \frac{I_i}{I_{ref}}=10 [/tex]
therefore, we can rewrite the original equation as:
[tex]m_i-5 = -2.5 \log_{10}(10)=-2.5[/tex]
And so, the magnitude of the star i is
[tex]m_i = 5-2.5 = 2.5[/tex]