Question

Log10^2=0.3010and log10^3=0.4771 evaluate log10^45

Answer 1

makes no sense.

log 10^2 = log 100 = 2 , not .3010
(if no base is shown, the base of the log expression is 10 by default)

Answer 2

If your question means:

log(base10) ( 2 ) = 0.30103

log(base10) ( 3 ) = 0.4771

evaluate log(base10) ( 4.5 )

then

4.5 = 9 / 2 = 3^2 / 2

log(base10) ( 4.5 ) =

2 • log(base10) ( 3 ) - log(base10) ( 2 ) =

2 • 0.4771 - 0.30103 = 0.65317

Answer 3

assuming base 10, with your unusual notation
(most would write log10(2) or log_10(2) since 10^2 = 10 to the second power)
log45 = log9 + log5 = 2log3 + log5 = 2log + log(10/2)
= 2log3 + log10 - log2
= 2*.4771 + 1 - 0.3010
= 1.6532