Exponential and Logarithmic Functions - Sketching Logarithmic Functions and Solving Equations Graphically

The logarithmic curve \(y=\ln(x-3)\) is defined only for \(x>3\); the stated positive-x condition alone is insufficient.

Sketch the graph of \(y=\ln \left( x-3 \right)\) for \(x>0\). Express \(\left( x-3 \right){{\mathrm{e}}^{3x}}=4\) in the form \(\ln \left( x-3 \right)=ax+b\) and state the value of \(a\) and of \(b\). Insert on your sketch the additional graph required to illustrate how a graphical solution of the equation \(\left( x-3 \right){{\mathrm{e}}^{3x}}=4\) may be obtained.