Expand the following logs:[tex]log_{7} \sqrt{a^{3}b^{9} }[/tex][tex]log_{6} (\frac{x^{5} }{y^{9} } )[/tex][tex]log_{8} (x^{3} y^{7} )[/tex]

QUESTION 1The given logarithm is[tex]\log_7\sqrt{a^3b^9}[/tex]We rewrite to obtain;[tex]\log_7(a^3b^9)^{\frac{1}{2}}[/tex]Use the power rule of logarithm ; [tex]\log_a(m^n)=n\log_a(m)[/tex][tex]\frac{1}{2}\log_7(a^3b^9)[/tex]Use the product rule; [tex]\log_a(mn)=\log_a(m)+\log_a(n)[/tex][tex]\frac{1}{2}[\log_7(a^3)+\log_7(b^9)][/tex]Use the power rule of logarithms again;[tex]\frac{1}{2}[3\log_7(a)+9\log_7(b)][/tex]Or[tex]\frac{3}{2}\log_7(a)+\frac{9}{2}\log_7(b)][/tex]QUESTION 2Given;[tex]\log_6(\frac{x^5}{y^9})[/tex]Apply the quotient rule of logarithm; [tex]\log_a(m)-\log_a(n)=\log_a(\frac{m}{n} )[/tex][tex]\log_6(\frac{x^5}{y^9})=\log_6(x^5)-\log_6(y^9)[/tex]Apply the power rule to get;[tex]\log_6(\frac{x^5}{y^9})=5\log_6(x)-9\log_6(y)[/tex]QUESTION 3Given;[tex]\log_8(x^3y^7)[/tex]Use the product rule to get;[tex]=\log_8(x^3)+\log_8(y^7)[/tex]Use the power rule now;[tex]=3\log_8(x)+7\log_8(y)[/tex]