The interval [2.5, ∞) exists where x ≥ 2.5 or x ≥ 2.5 or 2.5 ≤ x < ∞, which results in a colony having at least 320,000 bacteria.
The given exponential equation can be described as follows;
10,000 × 4ˣ ≥ 320,000
simplifying the above equation, we get
4ˣ ≥ 320,000/10,000
4ˣ ≥ 32
Taking log on both sides of the equation, we get
log (4ˣ) ≥ log (32)
simplifying the above equation, we get
x · log 4 ≥ log (32)
x ≥ log (32)/log 4
x ≥ 2.5
Therefore, the value of x ≥ 2.5.
The interval the colony will have at least 320,000 bacteria exists where x ≥ 2.5 or 2.5 ≤ x < ∞ which gives, the interval [2.5, ∞).
Therefore, the correct answer is option a) [2.5, ∞).
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