Solution:
[tex]\text{First take the logarithm on both sides of the equation $2^a=3:$}[/tex]
[tex]\log(2^a)=\log3[/tex]
[tex]\text{Recall: $\boxed{\log_wx^y=y.\log_wx}$}[/tex]
[tex]a.\log2=\log3[/tex]
[tex]a=\dfrac{\log3}{\log2}[/tex]
[tex]\text{Next, take the logarithm on both sides of the equation $5^b=2:$}[/tex]
[tex]b.\log5=\log2\\[/tex]
[tex]b=\dfrac{\log2}{\log5}[/tex]
[tex]\text{Solve for $c$ in $abc=1:$}[/tex]
[tex]abc=1[/tex]
[tex]\dfrac{\log3}{\log2}\times\dfrac{\log2}{\log5}\times c=1[/tex]
[tex]c.\log3=\log5[/tex]
[tex]\log3^c=\log5[/tex]
[tex]\boxed{3^c=5}[/tex]
