The rules for combining logs are

1.logb(ac)=logb(a)+logb(c)2.logb(ac)=logb(a)−logb(c)3.logb(ar)=rlogb(a)

We see that:

log4(10)−3log4(36)+3log4(6)=log4(10)−log4(363)+log4(63) using 3.=log4(10)−log4(46656)+log4(216)=log4(10×216)−log4(46656) using 1.=log4(10×21646656) using 2.=log4(216046656)=log4(5108) on cancelling common factors.

Hence c=5108.

To calculate log4(5108) to 4 decimal places we use the fact that for any positive base b:

logb(c)=ln(c)ln(b)=log10(c)log10(b)

and we can use either of the log functions, ln or log10 on our calculators to find the value.

Using ln we find:

log4(5108)=ln(5108)ln(4)=−2.2165