Exponential and logarithmic inequalities

When does a decaying model exceed a threshold: before or after crossing it? The same inequality sign gives opposite time directions for growth and decay. Use exponential functions, inequalities, logarithms and equations.

$2^t\ge8\iff t\ge3;\qquad (1/2)^t\ge1/8\iff t\le3.$

For a>1, aˣ increases and preserves exponent order. For 0<a<1, it decreases and reverses order. The base is positive: this is monotonicity, not multiplication by a negative number. If t is time after starting, add t≥0: answers [3,∞) and [0,3].

Growth

2ᵗ≥8

Decay

2⁻ᵗ≥1/8

Increasing 2ᵗ≥8 gives t≥3; decreasing 2⁻ᵗ≥1/8 gives t≤3. Separate vertical scales; hours and hypothetical values. Both cross at t=3 with the same comparison sign.

Domain before logarithm comparisons

logₐx has the same base-dependent monotonicity. First require x−1>0; then compare the argument with $(1/2)^{-2}=4$ :

$\log_{1/2}(x-1)>-2\iff0<x-1<4\iff1<x<5.$

x=1 is undefined; x=5 gives equality, excluded by the strict sign. OpenStax

“And” means intersection

$2^t\ge4,\quad(1/2)^t>1/16\iff2\le t<4.$

t=2 gives an allowed equality in the first condition; t=4 gives a forbidden equality in the second. Do not take the union.

Substitute and return

$4^t-5\cdot2^t+4\le0;\quad u=2^t>0;\quad(u-1)(u-4)\le0\iff1\le u\le4\iff0\le t\le2.$

Factoring gives an interval for u; the increasing function 2ᵗ returns the interval for t. Intersect with the original model domain. Base 1 and nonpositive bases are excluded here.