Logarithm question 2, 10 points

(log5 4 * log2 10) / log 25 √10
how do i simplify this without using calculator?
please show me the workings

formula : log(b) a = log(e) a/ log(e) b
using this to all terms in he equation,
(log4/log5 * log10/log2) / (log[root10]/log25)
{NOTE : log without base specified means log to the base e }
now log4 = 2*log2
so eqn becomes...
2log10/log5 * log 25/log[root10]
now log 25 = 2log5
log 10 = 2log[root10]
so answer is 8

You need to convert to a single base, using change of base formula:
log_b (x) = log_a (x) / log_a (b)

log₅(4) = log₁₀(4) / log₁₀(5)
. . . . . = log₁₀(2^2) / log₁₀(5)
. . . . . = 2 log₁₀(2) / log₁₀(5)

log₂(10) = log₁₀(10) / log₁₀(2)
. . . . . . . = 1 / log₁₀(2)

log₂₅(√10) = log₁₀(√10) / log₁₀(25)
. . . . . . . . = 1/2 log₁₀(10) / log₁₀(25)
. . . . . . . . = 1 / (2 log₁₀(25))
. . . . . . . . = 1 / (2 log₁₀(5^2))
. . . . . . . . = 1 / (4 log₁₀(5))


log₅(4) * log₂(10) / log₂₅(√10)

. . 2 log₁₀(2) . . . . . 1 . . . . . . . . 1
= -------------- * ------------ / ---------------
. . log₁₀(5) . . . log₁₀(2) . . . 4 log₁₀(5)

. . 2 log₁₀(2) . . . . . 1 . . . . . 4 log₁₀(5)
= -------------- * ------------ *
. . log₁₀(5) . . . log₁₀(2)

= 2 * 1 * 4

= 8

=[log 4/log 5 *log 10/log 2] / 1/2 log 10 /log 25=
=2.86135/(0.5 /log 25) =
=8
God bless you.