MATH SOLVE

5 months ago

Q:
# hello, Can someone please explain on how to do this? thank you!Graph the logarithmic function. y=log(x-3)

Accepted Solution

A:

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.

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.