Evaluate the logarithmic expression Log subscript5^25 +log subscript 5^625/logsubscript5^100. The value of the expression, rounded to the nearest hundredth, is____.

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03-07-2018
Mathematics

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Evaluate the logarithmic expression Log subscript5^25 +log subscript 5^625/logsubscript5^100. The value of the expression, rounded to the nearest hundredth, is____.

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iuliaelisa15 iuliaelisa15 · Answer 1 · 2018-07-03T13:53:19+02:00

Hello !
Here are some rules for logarithms :    log ₐ b = n ⇔ aⁿ = b
                                                             log ₐ aⁿ = n
                                                             log ₐ (b·c) = logₐ b + log ₐ c
                                                             log ₐ bⁿ = n· log ₐ b                                        ____________________________________________________
25=5²
625=5⁴
100 = 5² · 2²
____________________________________________________

(log₅ 25 + log₅ 625) / log₅100 =
(log₅ 5²+ log₅ 5⁴) / log₅ (5² · 2²) =
(log₅ 5²+ log₅ 5⁴) /( log₅ 5² + log₅ 2²) =
(2 + 4)/(2+log₅ 2² ) =
6/ (2+2·log₅2)=
6/2 + 6/ (2·log₅2) =

3+ 3/log₅2
log₅2 = 0.5

So 3+ 3/log₅2 = 3 + 3/0.5 =3 + 6 = 9

Answer : 9