Question

for what range of positive integers is log10 (x-40)+log(60-x)<2?plss I need an answer urgently

Answer 1

First of all consider the definition of logs
remember we can only take logs of positive numbers
so for
log( (x-40)+ log(60-x)) we could only use 40<x<60 or
else our logs would be undefined

use rules of logs

log( (x-40)(60-x) ) < 2
(x-40)(60-x) < 10^2
-x^2 + 20x - 2400 < 100
x^2 - 20x + 2500 > 0

since y = x^2 - 20x + 2500 lies entirely above the x axis
all values of x would satisfy this inequality.

so let's go back to our 40 < x < 60
let x = 40.001
is log(.001) + log 19.999 < 2 ? YES
let x = 50
is log10 + log10 < 2, NO, it is equal to 2
let x = 59.999
is log 19.999 + log .001 < 2 ? YES

mmmhh, my analysis shows that we could have
40 < x < 50 OR 50 < x < 60

that is, all values between 40 and 50 OR
all values between 50 and 60

Answer 2

18

Answer 3

amogus :)