How do you graph the following logs?
f(x)=log5 (x-2)
f(x)=log5 x-2
f(x)= log5 x
f(x)=log5 (x+2)
use f(x)= log5 x as your reference graph.
You should know what your standard log function graph looks like
If you translate the reference graph 2 units to the right, you get the first equation
If you translate the reference graph 2 units to the left, you get the last equation
If you translate the reference graph 2 units downwards, you get the second equation
To graph logarithmic functions like the ones you've provided, follow these steps:
1. Determine the domain: For a logarithmic function, the argument (the expression inside the logarithm) must be greater than zero. So, find the values of x that make the argument greater than zero.
2. Choose some values for x: Pick a few values for x from the determined domain. It's usually a good idea to select values that are spread out and include critical points like x-intercepts or asymptotes.
3. Calculate the corresponding y-values: Plug each chosen x-value into the logarithmic function and evaluate it to obtain the corresponding y-value.
4. Plot the points: Plot the (x, y) coordinates on a graph.
5. Connect the points: Use a smooth, continuous curve to connect the plotted points. Keep in mind any vertical asymptotes or other characteristics specific to the given logarithmic function.
Let's now apply these steps to each logarithmic function you provided:
1. f(x) = log5(x-2):
- Determine the domain: For this function, x-2 > 0, which means x > 2.
- Choose values for x: Select a few x-values greater than 2, such as x = 3, 4, and 5.
- Calculate corresponding y-values: Evaluate each x-value by substituting it into the function: f(3) = log5(3-2) = log5(1) = 0, f(4) = log5(4-2) = log5(2) ≈ 0.4307, f(5) = log5(5-2) = log5(3) ≈ 0.6826.
- Plot the points: Plot the points (3, 0), (4, 0.4307), and (5, 0.6826).
- Connect the points with a smooth curve.
2. f(x) = log5x-2:
- Determine the domain: In this case, x - 2 can be negative or zero. So, there is no restriction on the domain.
- Choose values for x: Select several x-values, such as x = 1, 2, 3, and 4.
- Calculate corresponding y-values: Evaluate each x-value using the function: f(1) = log5(1-2) = log5(-1) is undefined, f(2) = log5(2-2) = log5(0) is undefined, f(3) = log5(3-2) = log5(1) = 0, f(4) = log5(4-2) = log5(2) ≈ 0.4307.
- Plot the points: Plot the points (3, 0) and (4, 0.4307).
- Since the function is undefined for x = 1 and x = 2, leave gaps in the graph at these points.
3. f(x) = log5x:
- Determine the domain: For this function, x must be greater than 0.
- Choose values for x: Select various x-values, such as x = 0.2, 0.5, 1, 2, and 5.
- Calculate corresponding y-values: Evaluate each x-value using the function: f(0.2) = log5(0.2) ≈ -0.4307, f(0.5) = log5(0.5) ≈ -0.8614, f(1) = log5(1) = 0, f(2) = log5(2) ≈ 0.4307, f(5) = log5(5) ≈ 1.4307.
- Plot the points: Plot the points (0.2, -0.4307), (0.5, -0.8614), (1, 0), (2, 0.4307), and (5, 1.4307).
- Connect the points with a smooth curve.
4. f(x) = log5(x+2):
- Determine the domain: In this case, x+2 must be greater than zero, which means x > -2.
- Choose values for x: Select a few x-values greater than -2, such as x = -1, 0, 1, and 2.
- Calculate corresponding y-values: Evaluate each x-value using the function: f(-1) = log5(-1+2) = log5(1) = 0, f(0) = log5(0+2) = log5(2) ≈ 0.4307, f(1) = log5(1+2) = log5(3) ≈ 0.6826, f(2) = log5(2+2) = log5(4) ≈ 0.8614.
- Plot the points: Plot the points (-1, 0), (0, 0.4307), (1, 0.6826), and (2, 0.8614).
- Connect the points with a smooth curve.
Remember, these steps are general guidelines, and depending on the context or specific instructions, there may be additional considerations.