log85=b
using the base change rule
log85=log45/log48
log48 can be solved:
log48=x
4x=8
22x=23
2x=3
x=3/2
Therefore we can write:
log85=log45/(3/2)=b
log45=(3/2)b [1]
To make log45 into log410 we can use the product rule:
log42+log45=log4(2x5)=log410
So by adding log42 on both sides of equation [1] we can write
log42+log45=log42+ (3/2)b
log410=log42+ (3/2)b
But log42 can be solved:
log42=x
4x=2
22x=2
2x=1
x=1/2
Therefore we can conclude that
log410=1/2+ (3/2)b
log410=(1+3b)/2