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.
Here you change the base to see the curve flatten, then view the pair mirrored about y = x.
Shape — How the Base Changes the Curve
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Vary the base a to inspect y = log_a(x)
2
💡 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)
Symmetry with the Exponential
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y = 2^x and y = log₂x are symmetric about y = x
🔗 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 = loga x (a > 0, a ≠ 1, x > 0)
Domain: x > 0; range: all real numbers
Properties
loga 1 = 0, loga 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
loga f(x) = loga 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 = loga 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.
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Use the definition to rewrite it in exponential form (argument = basevalue).
x - 1 = 23 = 8
2
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 = 2x symmetric?
1
y = log2 x is the inverse of y = 2x.
y = 2x ⟺ x = log2 y
2
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.
2023 CSAT Math type, adapted
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
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Since base 3 > 1 the function increases (inequality kept); also apply the domain.