For y=kln(ax+b), where k,a are non-zero integers, the argument requires ax+b>0. The boundary x=- ba is a vertical asymptote. The full-domain range is R. The natural logarithm sends positive arguments close to zero to very negative values and grows without bound for large positive arguments. A negative k reverses these vertical directions. Method: solve ax+b>0 before drawing; find the zero from ax+b=1; find a y-intercept only if b>0; decide which side of the asymptote is allowed; sketch the branch and label all requested features. For y=-ln(x-2), the domain is x>2, the zero is x=3, and there is no y-intercept. The curve decreases from large positive values near x=2. Exam check: dividing an inequality by a negative inner coefficient reverses its direction. If a<0, the graph lies to the left of the asymptote. A zero logarithm has argument 1, not argument 0.

For \(y=k\ln(ax+b)\), where \(k,a\) are non-zero integers, the argument requires \(ax+b>0\). The boundary \(x=-\dfrac ba\) is a vertical asymptote. The full-domain range is \(\mathbb R\).

The natural logarithm sends positive arguments close to zero to very negative values and grows without bound for large positive arguments. A negative \(k\) reverses these vertical directions.

Method: solve \(ax+b>0\) before drawing; find the zero from \(ax+b=1\); find a \(y\)-intercept only if \(b>0\); decide which side of the asymptote is allowed; sketch the branch and label all requested features.

For \(y=-\ln(x-2)\), the domain is \(x>2\), the zero is \(x=3\), and there is no \(y\)-intercept. The curve decreases from large positive values near \(x=2\).

Exam check: dividing an inequality by a negative inner coefficient reverses its direction. If \(a<0\), the graph lies to the left of the asymptote. A zero logarithm has argument \(1\), not argument \(0\).

Logarithmic graph domains and orientation

Logarithmic graph domains and orientation