AP Precalculus 2.11 Exponential & Logarithmic Equations — Notes

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General Form & Key Facts

What every logarithmic function looks like

f(x) = a · logᵦ x, b > 0

General form. a ≠ 0 and b ≠ 1, b > 0. Logarithmic and exponential functions are inverses of each other.

Domain: (0, ∞)

Inputs must be strictly positive. You cannot take a log of zero or a negative number. The y-axis (x = 0) is a vertical asymptote.

Range: (−∞, ∞)

Logarithmic functions can produce any real number output — unbounded in both directions.

Vertical asymptote: x = 0

Since domain is (0,∞), the function approaches ±∞ as x → 0⁺ from the right. Left end behavior uses x → 0⁺.

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Logs are inverses of exponentials

Because f(x) = logᵦ x is the inverse of f(x) = bˣ, their graphs are reflections over y = x. Every property of log functions mirrors what you know about exponentials — domain/range swap, graphs reflect, limits swap.

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Always Increasing or Decreasing · Always One Concavity

No local extrema · No inflection points

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Two "always" rules — just like exponentials

Always increasing OR always decreasing — never switches direction → no relative (local) extrema (unless on a closed interval).

Always concave up OR always concave down — never switches concavity → no points of inflection.

Condition	Direction	Concavity	lim x→0⁺	lim x→+∞
a > 0, b > 1	⬆️ Increasing	Concave Down	−∞	+∞
a < 0, b > 1	⬇️ Decreasing	Concave Up	+∞	−∞
a > 0, 0 < b < 1	⬇️ Decreasing	Concave Up	+∞	−∞
a < 0, 0 < b < 1	⬆️ Increasing	Concave Down	−∞	+∞

↔️

End Behavior — Example 1

Writing limit statements for logarithmic functions

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The two limit statements

For f(x) = a·logᵦ x, there are always two limits to write:

lim x→0⁺ f(x) — left end (approaching the vertical asymptote)
lim x→+∞ f(x) — right end (growing without bound)

Use the sign of a and whether b > 1 or 0 < b < 1 to determine ±∞.

a) f(x), a > 0, b > 1

lim x→0⁺ = −∞
lim x→+∞ = +∞

b) h(x), a < 0, b > 1

lim x→0⁺ = +∞
lim x→+∞ = −∞

c) g(x) = 2log₃x, a=2>0, b=3>1

lim x→0⁺ = −∞
lim x→+∞ = +∞

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Increasing, Decreasing & Concavity — Example 2

Read the sign of a and b from the graph or formula

a) a < 0, b > 1

Decreasing
Concave Up

b) a > 0, b > 1

Increasing
Concave Down

c) h(x) = −4log₆x

Decreasing
Concave Up

📌 How to determine direction and concavity from a formula

1

Check the sign of a (the coefficient). a > 0 and b > 1 → increasing, concave down.

2

a < 0 and b > 1 → decreasing, concave up. (Flipping a flips both direction and concavity.)

3

0 < b < 1 (fractional base) acts like a < 0 for direction — it flips the graph.

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Finding k from Tables — Example 3

Identify the pattern in x-values, then match to the log function

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The Strategy

Look at the x-values. They will be powers of the base b (possibly scaled or shifted). Find what the x-values are powers of → that tells you b. Then the missing k is whatever x gives the next output in the pattern.

f(x) = log₂x + 1

x	f(x)
1	1
2	2
k	3
8	4
16	5

x = 1, 2, k, 8, 16 are powers of 2
1=2⁰, 2=2¹, k=2²=4

g(x) = 5log₃(x/2)

x	g(x)
k	0
6	5
18	10
54	15
162	20

x/2 values: 3⁰·1, 3¹·1… so x values are multiples of 2
6=3¹·2, 18=3²·2, k=3⁰·2=2

h(x) = −10(log₂(x−3) − 1)

x	h(x)
4	10
5	0
7	−10
k	−20
19	−30

x−3 must be powers of 2
4−3=1=2⁰, 5−3=2=2¹, 7−3=4=2²
k−3=8=2³ → k=11

l(x) = (7/3)(5 + ln(x))

x	l(x)
e⁻²	7
e	14
k	21
e⁷	28
e¹⁰	35

Exponents: −2, 1, n, 7, 10
Pattern: p = −2 + 3n → n=2: p=4
k = e^(−2+3·2) = e⁴ → k=e⁴

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Domain of Logarithmic Functions — Example 4

The argument must be strictly greater than zero

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The Domain Rule

For f(x) = a·logᵦ(expression), set the argument > 0 and solve. The range is always (−∞, ∞) — unchanged by any transformation of the argument.

f(x) = 2log₃x

Argument: x
Set x > 0

Domain: (0, ∞)
Range: (−∞, ∞)

g(x) = −5log₂(x − 3)

Argument: x − 3
Set x − 3 > 0 → x > 3

Domain: (3, ∞)
Range: (−∞, ∞)

h(x) = 8log(2x + 3)

Argument: 2x + 3
Set 2x + 3 > 0 → x > −3/2

Domain: (−3/2, ∞)
Range: (−∞, ∞)

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Quick Reference

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🔑 Key Facts

f(x) = a·logᵦ x,  b>0, b≠1

General form — a≠0

Domain: (0,∞) · Range: (−∞,∞)

Always — input must be positive

a>0, b>1 → Inc, Concave Down

a<0, b>1 → Dec, Concave Up

Domain rule: argument > 0

⚠️ Common Mistakes

❌ Left end behavior: x→−∞

Log functions have domain (0,∞). Left end is x→0⁺, NOT x→−∞.

❌ a>0 always means concave up

For logs, a>0 means concave down. (Opposite of exponentials — they're inverses!)

❌ Range is restricted like domain

Domain is (0,∞) but range is always (−∞,∞) — logs can output any real number.

❌ Logs have inflection points

Like exponentials, logs have ONE fixed concavity — no inflection points ever.

LogarithmicFunctions

Logs are inverses of exponentials

Two "always" rules — just like exponentials

The two limit statements

The Strategy

The Domain Rule

Logarithmic
Functions