Number of integers satsifying inqualities with logarithm I am trying to solve the problem of finding the integers x satisfying the inequalities: 2<log_x 45<3 I realize this is a very basic question on logarithms and I have the key with the answers 4, 5, 6 and have confirmed this is correct using Wolfram Alpha but which steps should I take to reach that answer?

Sanai Ball

Sanai Ball

Answered question

2022-10-05

Number of integers satsifying inqualities with logarithm
I am trying to solve the problem of finding the integers x satisfying the inequalities:
2 < l o g x 45 < 3
I realize this is a very basic question on logarithms and I have the key with the answers 4, 5, 6 and have confirmed this is correct using Wolfram Alpha but which steps should I take to reach that answer?

Answer & Explanation

Kelly Ibarra

Kelly Ibarra

Beginner2022-10-06Added 6 answers

You can rewrite this as
2 < ln ( 45 ) / ln ( x ) < 3
and then
2 ln ( x ) < ln ( 45 ) < 3 ln ( x )
since ln ( x ) must be positive. Then this becomes
ln ( x 2 ) < ln ( 45 ) < ln ( x 3 )
and finally exponentiating turns this into the inequality:
x 2 < 45 < x 3 .
Now just plug in a few integers to figure out exactly which ones work.

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