high school Logarithmic Function

Logarithmic Function

The logarithm y = log_a x is the mirror twin of the exponential y = a^x. The exponential gives the result of multiplying so many times; the logarithm asks the reverse, how many times to multiply to reach it, so their graphs fold across y = x. So it always passes through (1, 0), and as x nears 0 it dives down with the y-axis as its asymptote. Change the base and the curve flattens, and the pair sits mirrored about y = x.

Shape — How the Base Changes the Curve

Base a2

💡 Every log curve passes through (1, 0)

①log_a(1) = 0 → y = 0 at x = 1 regardless of the base

②Larger base → flatter curve

③x → 0⁺ ⇒ y → -∞ (y-axis is the asymptote)

④This sketch uses a ≥ 1.5. Decay for 0 < a < 1 is the sentence in the exam box

Symmetry with the Exponential

🔗 Inverse Function Relationship

①y = log_a(x) is the inverse of y = a^x

②Graphs are reflections across y = x

③Domain and range swap

④(0, 1) ↔ (1, 0)

Properties of the Log Function

Definition

y = log_a x (a > 0, a ≠ 1, x > 0)

Domain: x > 0; range: all real numbers

Properties

log_a 1 = 0, log_a a = 1

Passes through (1, 0) and (a, 1)

📐 Monotonicity & Asymptote

①a > 1: increasing

②0 < a < 1: decreasing

③Asymptote: y-axis (x = 0)

④Not defined for x ≤ 0

Log Equations & Inequalities

Log Equation

log_a f(x) = log_a g(x) ⟹ f(x) = g(x)

Same base ⇒ compare arguments (and check argument > 0)

⚠️ Key Idea for Log Inequalities

①a > 1: log_a M > log_a N ⟺ M > N (preserved)

②0 < a < 1: log_a M > log_a N ⟺ M < N (flipped!)

③Always verify argument > 0

Wrap-up

Log Core

y = log_a x: domain (0, ∞), range ℝ, asymptote x = 0

Inverse of the exponential; symmetric across y = x

🎯 Exam Points

①Passes through (1, 0): log_a 1 = 0

②a > 1 increases, 0 < a < 1 decreases

③Inverse: y = a^x ↔ y = log_a x

④Log equations: equalize bases, then compare arguments

⑤Always check argument > 0

Worked Examples & Past Exam

Example 1

Solve the logarithmic equation log₂ (x − 1) = 3.

Use the definition to rewrite it in exponential form (argument = base^value).

x - 1 = 2³ = 8

Solve for x and check the domain x − 1 > 0.

x = 9 (x - 1 = 8 > 0)

▸ x = 9

Convert the log equation to exponential form, then always check the argument is positive.

Example 2

About which line are the graphs of y = log₂ x and y = 2^x symmetric?

y = log₂ x is the inverse of y = 2^x.

y = 2^x ⟺ x = log₂ y

Graphs of inverse functions are symmetric about the line y = x.

axis of symmetry: y = x

▸ Symmetric about the line y = x

Exponential and logarithmic functions are inverses, so they are symmetric about y = x.

exam-style

Solve the logarithmic inequality log₃ (x − 2) < 2 (include the domain).

①2 < x < 11

②x < 11

③x > 2

④0 < x < 11

⑤2 < x < 9

▸ ① 2 < x < 11

Since base 3 > 1 the function increases (inequality kept); also apply the domain.

log₃ (x-2) < 2 ⟺ 0 < x - 2 < 3²

Solve the inequality for the range.

2 < x < 11

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Exponential Function

Common Logarithm

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