Question

given f(x)=ln(x-1), find the equation of the tangent to f(x) at x=3, find the zeroes of f(x) analytically, sketch the graphs of f(x) and the tangent line on the same axes, write your procedure for finding the zeroes of f(x) graphically. please show all steps!

Answer 1

\[f(x)=\ln(x-1)=0\implies x-1=1\implies x = 2\]

Answer 2

the point
\[(3,\ln(2))\] is on the graph and to find the slope, find
\[f'(x)=\frac{1}{x-1}\]
\[f'(3)=\frac{1}{2}\] use point slope formula with
\[m=\frac{1}{2},x_1=3,y_1=\ln(2)\] do find the equation for the tangent line

Answer 3

can you explain wat does the 'ln' mean?

Answer 4

how do you find the zeroes analytically??