Problem

Analyze \(f(x)~=~\log_{5}(x~-~2)~-~1\). State the base and argument condition; give the domain, range, asymptote, intercepts, reflection, and monotonic behavior; and map the exact anchors for arguments \(1\), \(5\), and \(1/5\).

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

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Base and argument: base 5; x − 2 > 0.
Features: domain x < 2; range all real numbers; asymptote x = 2; x-intercept (7, 0); no y-intercept; no reflection; increasing.
Anchors by argument: 1 → (3, −1), 5 → (7, 0), 1/5 → (11/5, −2).

Base and argument: base 5; x − 2 > 0.
Features: domain x > 2; range all real numbers; asymptote x = 2; x-intercept (7, 0); no y-intercept; no reflection; increasing.
Anchors by argument: 1 → (3, −1), 5 → (7, 0), 1/5 → (11/5, −2).

Base and argument: base 5; x − 2 > 0.
Features: domain x > 2; range all real numbers; asymptote x = 2; x-intercept (5, 0); no y-intercept; no reflection; increasing.
Anchors by argument: 1 → (3, −1), 5 → (5, 0), 1/5 → (11/5, −2).

Base and argument: base 5; x − 2 > 0.
Features: domain x > 2; range all real numbers; asymptote x = 2; x-intercept (7, 0); y-intercept (0, −1); no reflection; increasing.
Anchors by argument: 1 → (3, −1), 5 → (7, 0), 1/5 → (11/5, −2).

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Analyze \(f(x)~=~\log_{5}(x~-~2)~-~1\). State the base and argument condition; give the domain, range, asymptote, intercepts, reflection, and monotonic behavior; and map the exact anchors for arguments \(1\), \(5\), and \(1/5\).

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