Exponential & Log Equations
Solve exactly: 3^(x − 1) = 81
x = 5
Let f(x) = 3ˣ + 1. State the range and horizontal asymptote of f.
Range: y > 1
Horizontal asymptote: y = 1
A substance has a half-life of 4 days. If 200 grams are present now, how much remains after 8 days?
50 grams
Solve exactly: log₂(x − 1) + log₂(x + 1) = 3. State the domain restriction you use.
x = 3 (domain: x > 1)
Let f(x) = 2ˣ⁻¹ + 3. Find f⁻¹(x) and evaluate f⁻¹(7).
f⁻¹(x) = log₂(x − 3) + 1
f⁻¹(7) = 3
A substance decays exponentially with a half-life of 6 days. After 6 days, 45 grams remain. Find the initial amount and write M(t).
Initial amount = 90 g
M(t) = 90(1/2)^(t/6)
Solve exactly: log₂(x) + log₂(x − 6) = 4. Show why one algebraic solution must be rejected.
x = 8
(x = −2 is rejected. log₂ requires a positive input)
Let f(x) = 5ˣ⁺² − 4. Find the range, horizontal asymptote, and a formula for f⁻¹(x), stating its domain.
Range: y > −4
Asymptote: y = −4
f⁻¹(x) = log₅(x + 4) − 2
Domain x > −4
Two samples start with equal mass. Sample A has a half-life of 2 days; Sample B has a half-life of 5 days. After 10 days, what fraction of each remains, using exact fractions?
Sample A: (1/2)^5 = 1/32 remains
Sample B: (1/2)^2 = 1/4 remains
Sample B has more mass left