log8(5) = b. Express log4(10) in terms of b

log₈5=b

using the base change rule

log₈5=log₄5/log₄8

log₄8 can be solved:

log₄8=x

4^x=8

2^2x=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

4^x=2

2^2x=2

2x=1

x=1/2

Therefore we can conclude that

log₄10=1/2+ (3/2)b 

log₄10=(1+3b)/2