Problem

Analyze \(f(x)~=~\log_{2}(x)\). State the base and argument condition; give the domain, range, asymptote, intercepts, reflection, and monotonic behavior; and map the exact anchors for arguments \(1\), \(2\), and \(1/2\).

Prompt coordinate grid scaled for the stated logarithmic equation; the curve, anchors, and asymptote are withheld.

Big Picture

What this problem is really about

Begin with the logarithm’s argument restriction, because its boundary gives the vertical asymptote and determines whether a y-intercept can exist. Find the x-intercept by setting the output to zero. For exact anchors, use arguments equal to one, the base, and the reciprocal of the base, then connect them on the allowed side of the asymptote.

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Preview only

Base and argument: base 2; x > 0.
Features: domain x < 0; range all real numbers; asymptote x = 0; x-intercept (1, 0); no y-intercept; no reflection; increasing.
Anchors by argument: 1 → (1, 0), 2 → (2, 1), 1/2 → (1/2, −1).

Base and argument: base 2; x > 0.
Features: domain x > 0; range all real numbers; asymptote x = 0; x-intercept (0, 1); no y-intercept; no reflection; increasing.
Anchors by argument: 1 → (0, 1), 2 → (2, 1), 1/2 → (1/2, −1).

Base and argument: base 2; x > 0.
Features: domain x > 0; range all real numbers; asymptote x = 0; x-intercept (1, 0); y-intercept (0, 0); no reflection; increasing.
Anchors by argument: 1 → (1, 0), 2 → (2, 1), 1/2 → (1/2, −1).

Base and argument: base 2; x > 0.
Features: domain x > 0; range all real numbers; asymptote x = 0; x-intercept (1, 0); no y-intercept; no reflection; increasing.
Anchors by argument: 1 → (1, 0), 2 → (2, 1), 1/2 → (1/2, −1).

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Analyze \(f(x)~=~\log_{2}(x)\). State the base and argument condition; give the domain, range, asymptote, intercepts, reflection, and monotonic behavior; and map the exact anchors for arguments \(1\), \(2\), and \(1/2\).

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