Exponential and logarithmic functions flashcards

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The number e
The irrational number 2.71828… for which the gradient of y = eˣ equals its y-value at every point.
Natural logarithm ln x
log to base e: ln x is the power e must be raised to in order to give x. Defined only for x > 0.
Asymptote
A line that a curve approaches more and more closely but never meets, e.g. y = c for y = e^(ax+b) + c.
Exponential growth / decay model
A model y = Ae^(kt): growth if k > 0, decay if k < 0. Its rate of change is proportional to its current value.
Initial value of a model
The value when t = 0; for y = L + Be^(−kt) it is L + B, since e⁰ = 1.
Long-term value of y = L + Be^(−kt), k > 0
L: as t → ∞, e^(−kt) → 0. State it in context.
Inverse relationship between e and ln
e^(ln x) = x for x > 0, and ln(eˣ) = x for all real x. (No — learn it)
Derivative of e^(kx)
d/dx(e^(kx)) = ke^(kx); in particular d/dx(eˣ) = eˣ. (No — learn it)
aˣ in terms of e
aˣ = e^(x ln a). (Yes)
Derivative of aˣ
d/dx(aˣ) = aˣ ln a, from aˣ = e^(x ln a). (No — derive it)
Linear form of y = axⁿ
log y = log a + n log x: plot log y against log x; gradient n, intercept log a. (No — derive it)
Linear form of y = kbˣ
log y = log k + x log b: plot log y against x; gradient log b, intercept log k. (No — derive it)
Solution of e^(ax+b) = p (p > 0)
x = (ln p − b)/a. No solutions if p ≤ 0. (No — derive it)
Key features of y = eˣ
Passes through (0, 1); always positive; increasing; x-axis is an asymptote as x → −∞.
How are the graphs of y = eˣ and y = ln x related?
Each is the reflection of the other in y = x; (0, 1) ↔ (1, 0); horizontal asymptote ↔ vertical asymptote.
Asymptote and range of y = e^(ax+b) + c
Asymptote y = c; range y > c (y < c if the exponential term is multiplied by a negative number).
Vertical asymptote of y = ln(ax + b) + c
x = −b/a, where the bracket is zero; the curve exists only where ax + b > 0.
Shape of a cooling curve T = L + Be^(−kt)
Starts at L + B, falls fastest at first, then flattens towards the asymptote T = L.
Data give a straight line of log y against log x. Which model?
A power law y = axⁿ.
Data give a straight line of log y against x. Which model?
An exponential y = kbˣ.
Why does eˣ = −3 have no solutions?
eˣ > 0 for every real x, so it can never equal a negative number (and ln of a negative is undefined).
Why must you isolate eˣ before taking ln?
ln(A + B) ≠ ln A + ln B, so ln can only be applied to a single exponential (or product), not a sum.
What is the meaning of b in y = kbˣ?
The factor y is multiplied by when x increases by 1: b > 1 growth, 0 < b < 1 decay.
Give a typical limitation of an exponential growth model.
It predicts unlimited growth, whereas real quantities are limited (space, food, market size).
How might a decay model to zero be refined if there is a minimum value S?
Add a constant: V = S + Be^(−kt), which tends to S instead of 0.
Solve e^(2x − 5) = 3 exactly.
2x − 5 = ln 3, so x = (5 + ln 3)/2.
Solve ln(2x + 1) = 4 exactly.
2x + 1 = e⁴, so x = (e⁴ − 1)/2.
Solve e^(2x) − 5eˣ + 6 = 0.
Let u = eˣ: (u − 2)(u − 3) = 0, so x = ln 2 or x = ln 3.
y-intercept and asymptote of y = 2e^(−x) + 3?
(0, 5); asymptote y = 3 (approached as x → ∞); range y > 3.
N = 200e^(0.05t). When does N double?
e^(0.05t) = 2, so t = ln 2 / 0.05 ≈ 13.9.
Gradient of y = 5ˣ at x = 2?
dy/dx = 5ˣ ln 5, so 25 ln 5 (≈ 40.2).
log₁₀ y against log₁₀ x is a line through (0.2, 0.5) and (1.4, 2.9). Find a and n in y = axⁿ.
n = 2.4/1.2 = 2; intercept 0.5 − 2 × 0.2 = 0.1, so a = 10^0.1 ≈ 1.26.
ln y = 1.2 + 0.4x. Find k and b in y = kbˣ.
k = e^1.2 ≈ 3.32; b = e^0.4 ≈ 1.49.
T = 20 + 60e^(−0.1t). Find dT/dt at t = 0 and interpret.
dT/dt = −6e^(−0.1t) = −6: the temperature is falling at 6 degrees per unit time.
First step in solving 3e^(2x) + 1 = 13
Isolate: e^(2x) = 4, then take ln: 2x = ln 4, x = ln 2.
Undoing the intercept of a log graph
If the axis is log₁₀ y, use 10^( ); if it is ln y, use e^( ). Check before you calculate.
“Give an exact answer” in an e/ln question
Leave ln and e in the answer and simplify with the log laws (e.g. ln 4 = 2 ln 2); no decimals.
A substitution u = eˣ gives u = −2
Reject it, stating eˣ > 0 for all real x.
Differentiating 3ˣ with the power rule (x·3^(x−1))
Wrong: x is in the exponent. d/dx(3ˣ) = 3ˣ ln 3.

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