Through (1, 0), hugging the y-axis: The logarithm y = logₐx (a > 1) passes through (1, 0) (because logₐ1 = 0), rises slowly, and has the vertical asymptote x = 0 — it's only defined for x > 0.
IB-style question — key points of log
State the x-intercept and vertical asymptote of y = log x.
Step by step
- x-intercept: y = 0.
- As x → 0⁺, log x → −∞.
Final answer
x-intercept (1, 0); vertical asymptote x = 0.
You can't log zero or a negative: logₐx is only defined for x > 0. There's no y-intercept (x = 0 isn't allowed).
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Reflections in y = x: y = logₐx is the inverse of y = aˣ — each is the other reflected in the line y = x. So their key features swap: aˣ goes through (0, 1) with asymptote y = 0; logₐx goes through (1, 0) with asymptote x = 0.
y = aˣ
- through (0, 1)
- asymptote y = 0
- domain all x, range y > 0
y = logₐx (its inverse)
- through (1, 0)
- asymptote x = 0
- domain x > 0, range all y
Domain ↔ range swap: Because they're inverses, the domain of the log is the range of the exponential (x > 0), and vice versa.
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Asymptote x = 0, x-intercept (1, 0): To sketch y = logₐx (a > 1): draw the vertical asymptote x = 0, mark the x-intercept (1, 0), and curve upward to the right (slowly increasing, defined only for x > 0).
IB-style question — domain & range
State the domain and range of y = log x.
Step by step
- Only positive inputs allowed.
- Outputs cover every real value.
Final answer
Domain x > 0; range y ∈ ℝ.
It increases forever, slowly: log x keeps rising as x grows, but ever more slowly — there's no horizontal asymptote.
The inside shifts the asymptote: y = logₐ(x − h) + k shifts the graph h right and k up. The vertical asymptote moves to x = h, and the domain becomes x > h (the inside must stay positive).
IB-style question — shifted log
State the vertical asymptote and domain of y = log(x − 2).
Step by step
- Inside must be positive.
- Asymptote where the inside is 0.
Final answer
Vertical asymptote x = 2; domain x > 2.
Domain follows the asymptote: For logₐ(x − h), the domain is x > h — everything to the right of the vertical asymptote.