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Topic 2.11 · General Form
What is the general form of a logarithmic function?
f(x) = a · logᵦ x, b > 0
What are the restrictions on a and b?
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Topic 2.11 · Domain
What is the domain of f(x) = a·logᵦ x? Why?
Can you take log of 0 or a negative?
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Topic 2.11 · Range
What is the range of f(x) = a·logᵦ x?
Unlike the domain, the range has no restriction
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Topic 2.11 · Vertical Asymptote
Where is the vertical asymptote of f(x) = a·logᵦ x? How does it affect end behavior notation?
Left end behavior uses x → ?
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Topic 2.11 · Always Rules
Name the two 'always' rules for logarithmic functions. What do they imply?
No extrema · No inflection points
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Topic 2.11 · Behavior Table
a > 0 and b > 1 → increasing or decreasing? Concave up or down?
f(x) = a·logᵦ x
Think y = log x
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Topic 2.11 · Behavior Table
a < 0 and b > 1 → increasing or decreasing? Concave up or down?
h(x) = −4log₆x
Negative a flips everything
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Topic 2.11 · End Behavior
Write both limit statements for f(x) = 2log₃x (a > 0, b > 1).
f(x) = 2log₃x
Left: x→0⁺, Right: x→+∞
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✓ Domain
Domain: (0, ∞)
Argument must be strictly positive. log(0) and log(negative) are undefined. The y-axis x = 0 is a vertical asymptote.
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✓ General Form
f(x) = a·logᵦ x, b>0, a≠0, b≠1
a and b are constants. a ≠ 0, b ≠ 1, b > 0. Logs are the inverse of exponential functions.
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✓ Vertical Asymptote & Notation
x = 0 → left end: lim x→0⁺
The '+' superscript is required — you approach from the right side of 0. Writing x→−∞ is wrong because x can't be negative.
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✓ Range
Range: (−∞, ∞)
Logarithms can output any real number. The restricted domain does NOT restrict the range.
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✓ a>0, b>1
Increasing · Concave Down
a > 0, b > 1 → increasing and concave down. Think y = log x — rises but flattens (concave down). No extrema, no inflection.
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✓ Two 'Always' Rules
Always Inc/Dec · Always one concavity
Always increasing OR always decreasing → no local extrema.
Always concave up OR always concave down → no inflection points.
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✓ End Behavior: 2log₃x
lim x→0⁺ = −∞ · lim x→+∞ = +∞
a = 2 > 0, b = 3 > 1 → increasing → drops to −∞ near 0 and rises to +∞. Both limits are required on the AP exam.
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✓ a<0, b>1
Decreasing · Concave Up
a < 0, b > 1 → decreasing and concave up. Negative a flips both direction and concavity. Example: h(x)=−4log₆x.
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Topic 2.11 · Domain with Shift
Find the domain of g(x) = −5log₂(x − 3).
g(x) = −5log₂(x − 3)
Set the argument > 0
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Topic 2.11 · Finding k
x values 1, 2, k, 8, 16 follow a pattern. Find k.
f(x) = log₂x + 1 · f(k)=3
All x-values are powers of 2
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✓ Finding k from table
k = 4
1=2⁰, 2=2¹, k=2², 8=2³, 16=2⁴.
k = 2² = 4. Strategy: find what base the x-values are powers of.
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✓ Domain with Shift
Domain: (3, ∞)
Set x−3 > 0 → x > 3.
Domain: (3, ∞). The vertical asymptote shifts to x = 3. Range is still (−∞, ∞).