Use the properties of logarithms to find the exact values of

Use the properties of logarithms to find the exact values of the expressions. Do not use a calculator.

a) 10^(log30 -log5)

Use the log property : logA -logB = log(A/B)

So, 10^(log30/5 ) = 10^(log6)

Let y = 10^(log6) ; log y = log[10^(log6)]

logy = log6*log10

we know log10 = 1

SO, logy = log6

Hence y = 6

So, 10^(log30 -log5) =6

b) [log3(30)]*[ log30(9) ]

Use the property : loga(b) = logb/loga

[log(30)/log3]*[log(9)/log(30) ]
-- log30 gets cancelled

log(9)/log(3)

= 2log(3)/log(3)

= 2