Find all solutions from this logarithmic inequality: log(x)(base 1-2x) < log(3-4x)(base 1-2x) ?

So you have x<3-4x which means x<.6

You also have to have positive numbers, so x>0 and 3-4x>0 or x<.75

It is also nonsensical to have a negative base, so 1-2x>0 or x<.5

combined, you have 0<x<.5 is your answer.

Technically you can have a negative base, but then the results are complex numbers.

The base of a logarithm must be positive, so

0 < 1-2x

2x < 1

x < 1/2

log₁₋₂ᵪ(x) < log₁₋₂ᵪ(3-4x)

log(x)/log(1-2x) < log(3-4x)/log(1-2x)

log(x) < log(3-4x)

x < 3-4x

5x < 3

x < 3/5

0 < x < 1/2

log(x)(base 1-2x)<log(3-4x)(base 1-2x)

by removing log from both the sides,(its property oflog function...)

x<3-4x

x<3/5

x<0.6

but x is not equal to 0 ornagative number...

therefor x belongs to (0,0.6)

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log(x)(base 1-2x) < log(3-4x)(base 1-2x)

log(x)/log(1-2x) < log(3-4x)/log(1-2x)

　0 < x

　0 < 1 - 2x .......x < 1/2

　x ≠ 1/2

　0 < 3 - 4x .......x < 3/4

　in all , 0 < x < 1/2

nultiply both sides by [log(1-2x)]²

log(x)•log(1-2x) < log(3-4x)•log(1-2x)

　log(1-2x)[log(3-4x) - log(x)] > 0

　log(1-2x)log[(3-4x)/x]> 0

log(1-2x) > 0 ...........and ....... log[(3-4x)/x] > 0

　1 - 2x > 1 ..............and ....... (3 - 4x)/x > 1

　x < 0 ............no solution.--------#1

log(1-2x) < 0 ......and ....... log[(3-4x)/x] < 0

　0 < 1 - 2x < 1 .........and ....... 0 < (3 - 4x)/x < 1

　0 < x < 1/2..............and　　　　　　　　ＥＤＩＴ

　　0 < x(3 - 4x) < x²

　　0 > 4x² - 3x .......0 > x(4x - 3) .....0 < x < 3/4

　　3x - 4x² < x²

　　0 < 5x² - 3x ....0 < x(5x - 3) ........x < 0 , 3/5 < 0

　　　　in all , 3/5 < x < 3/4

　　　　no solution.---------------#2

Then ,from #1 and #2 , This inequality has no solution.

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