Evaluate log3 108 + log3 100 -2 log3 20
Using the log rules:
log3 108 = log3 (3^3 * 4) = log3 3^3 + log3 4 = 3 + log3 4
log3 100 = log3 (10^2) = log3 10^2 = 2 log3 10
log3 20 = log3 (2^2 * 5) = log3 2^2 + log3 5 = 2 + log3 5
Substituting in the values:
log3 108 + log3 100 -2 log3 20
= (3 + log3 4) + 2 log3 10 - 2(2 + log3 5)
= 3 + log3 4 + 2 log3 10 - 4 - 2 log3 5
= -1 + log3 4 + 2 log3 10 - 2 log3 5
Simplifying:
log3 4 = 2
log3 10 = log3 (2*5) = 1 + log3 5
Substituting in the values again:
-1 + 2 + 2(1 + log3 5) - 2 log3 5
= 3 + 2 log3 5 - 2 log3 5
= 3
Therefore, log3 108 + log3 100 -2 log3 20 = 3.