sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to solve the following inequality such that:

`(2/3)log_(1/4)(x + 2)^2 - 3 < = log_(1/4)(x + 5)^3 + log_(1/4)(x - 4)^3`

You need to use the following logarithmic identities, such that:

`n*log_a b = log_a b^n`

`log_a b + log_a c = log_a (b*c)`

`a = a*log_b b`

Reasoning by analogy yields:

`(2/3)log_(1/4)(x + 2)^2 = log_(1/4)((x + 2)^2)^(2/3)`

`log_(1/4)(x + 5)^3 + log_(1/4)(x - 4)^3 = log_(1/4)(x + 5)^3*(x - 4)^3`

`3 = 3*log_(1/4) (1/4) = log_(1/4) (1/4)^3`

`log_(1/4)((x + 2)^2)^(2/3) - log_(1/4) (1/4)^3 = log_(1/4) (((x + 2)^2)^(2/3))/((1/4)^3)`

Hence, you need to write the transformed inequality, such that:

`log_(1/4) (((x + 2)^2)^(2/3))/((1/4)^3) < = log_(1/4)(x + 5)^3*(x - 4)^3`

Since the base of logarithm is `1/4 < 1` , the inequality changes, such that:

`(((x + 2)^2)^(2/3))/((1/4)^3) >= (x + 5)^3*(x - 4)^3`

`4^3 root(3)((x+2)^4) >= (x + 5)^3*(x - 4)^3`

`root(3)((x+2)^4) >= (((x+5)(x-4))/4)^3`

You need to raise to cube both sides, such that:

`(x+2)^4 >= (((x+5)(x-4))/4)^9`

`(x+2)^4- (((x+5)(x-4))/4)^9 >= 0`

`(4^9(x + 2)^4 - (x+5)^9(x-4)^9)/4^9 >= 0`

Since `(x + 2)^4 > 0` for `x in R - {2}` , hence, `(4^9(x + 2)^4 - (x+5)^9(x-4)^9) > 0` for `x>4` .

Hence, solving the given logarithmic inequality, using the properties of logarithms, yields that the inequality holds for `x in (4,oo).`