PART A
For our beautiful sun, we'll have that:
[tex]b=1.4\cdot10^3[/tex]
This way,
[tex]\begin{gathered} M=-2.5\log (\frac{1.4\cdot10^3}{2.84\cdot10^{-8}}) \\ \\ \Rightarrow M=-9.74 \end{gathered}[/tex]
PART B
We'll have the equation:
[tex]-0.27=-2.5\log (\frac{b}{2.84\cdot10^{-8}})[/tex]
Solving for b,
[tex]\begin{gathered} -0.27=-2.5\log (\frac{b}{2.84\cdot10^{-8}})\rightarrow0.27=2.5\log (\frac{b}{2.84\cdot10^{-8}}) \\ \\ \rightarrow\frac{0.27}{2.5}=\log (\frac{b}{2.84\cdot10^{-8}})\rightarrow0.108=\log (\frac{b}{2.84\cdot10^{-8}}) \end{gathered}[/tex]
Now we'll use the following property:
[tex]c=\log _a(b)\Leftrightarrow b=a^c[/tex]
This way,
[tex]\begin{gathered} 0.108=\log (\frac{b}{2.84\cdot10^{-8}})\rightarrow e^{0.108}=\frac{b}{2.84\cdot10^{-8}} \\ \\ \Rightarrow b=2.84\cdot10^{-8}e^{0.108} \\ \\ \Rightarrow b=3.16\cdot10^{-8} \end{gathered}[/tex]
