The tutor demonstrates the useful change of base rule for logs.

Consider the problem

Simplify:log_√(27)⁵√(81)

This simplification might be suprisingly easy – if you know the change of base rule

log_ab=logb/loga

as well as the exponent to multiple log rule

loga^b=bloga

and the radical to exponent rule

Let’s proceed, starting with the original

log_√(27)⁵√(81)

which becomes, using change of base,

log⁵√(81)/log√(27)

Notice that 81=3^4, while 27=3^3. We can rewrite the expression as

log⁵√3^4}{log√3^3}\]

Next it becomes, by the radical to exponent rule,

log3^(4/5)/log3^(3/2)

Now, using the exponent to multiple rule, we arrive at

(4log3/5)/(3log3/2)

Cancelling out log3 from top and bottom, we have

(4/5)/(3/2) = (4/5)*(2/3) = 8/15

The reason the simplification works out so cleanly is that 81 and 27 both have the same base (in this case, 3).

HTH:)

Sources:

Travers, Kenneth, et al. Using Advanced Algebra. Toronto: Doubleday Canada Limited,   1977.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.