Steps:

1) determine the domain

2) determine the extreme limits of the function

3) determine critical points (where the derivative is zero)

4) determine the intercepts with the axis

5) do a table

6) put the data on a system of coordinates

7) graph: join the points with the best smooth curve

Solution:

1) domain

The logarithmic function is defined for positive real numbers, then you need to state x - 3 > 0

=> x > 3 <-------- domain

2) extreme limits of the function

Limit log (x - 3) when x → ∞ = ∞

Limit log (x - 3) when x → 3+ = - ∞ => the line x = 3 is a vertical asymptote

3) critical points

dy / dx = 0 => 1 / x - 3 which is never true, so there are not critical points (not relative maxima or minima)

4) determine the intercepts with the axis

x-intercept: y = 0 => log (x - 3) = 0 => x - 3 = 1 => x = 4

y-intercept: The function never intercepts the y-axis because x cannot not be 0.

5) do a table

x                          y = log (x - 3)

limit x → 3+            - ∞

3.000000001        log (3.000000001 -3) = -9

3.0001    log (3.0001 - 3) = - 4

3.1                       log (3.1 - 3) = - 1

4                          log (4 - 3) = 0

13                       log (13 - 3) = 1

103                     log (103 - 3) = 10

lim x → ∞             ∞

Now, with all that information you can graph the function: put the data on the coordinate system and join the points with a smooth curve.

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