I need to justify how a base in a logarithmic function graph is a base. Here are the choices:

The graph of f(x)= log_3 x+c must intersect with the line y= c+1 when x= 3.

The graph of f(x)= log_3 x+c must intersect with the line y= c when x= 3.

The graph of f(x)= log_3 x+c must intersect with the line x= c when y= 3.

The graph of f(x)= log_3 x+c must intersect with the line x= c+1 when y= 3.

Is it B? I'll be honest, I have no idea how to get the answer and the lesson didn't cover this.

you know that

log_3(3)=1

so, log_3(x)+c = 1+c when x=3

To justify which choice is correct, we need to understand the properties of logarithmic functions.

A logarithmic function with base b is defined as f(x) = log_b(x). The base b determines the behavior of the logarithmic function and the shape of its graph.

In the given options, we have f(x) = log_3(x) + c. This means that the base of the logarithmic function is 3.

To determine which option is correct, we can use the properties of logarithmic functions.

The graph of a logarithmic function f(x) = log_b(x) has a vertical asymptote at x = 0 and passes through the point (1, 0). The graph also has a reflection symmetry about the vertical line x = 1.

Now let's analyze each option:

A) "The graph of f(x) = log_3(x) + c must intersect with the line y = c + 1 when x = 3."
If we set x = 3, we have f(3) = log_3(3) + c = 1 + c.
So, the point of intersection would be (3, 1 + c). This option is not necessarily true because c can take any value, and the intersection point would vary.

B) "The graph of f(x) = log_3(x) + c must intersect with the line y = c when x = 3."
If we set x = 3, we have f(3) = log_3(3) + c = 1 + c.
So, the point of intersection would be (3, 1 + c). This option is correct because for any value of c, the logarithmic function will intersect with the line y = c at the point (3, 1 + c).

C) "The graph of f(x) = log_3(x) + c must intersect with the line x = c when y = 3."
If we set y = 3, we have 3 = log_3(x) + c.
Simplifying, we get log_3(x) = 3 - c.
This means that the point of intersection between the graph and the line x = c would occur when log_3(x) = 3 - c.
However, this does not necessarily happen at x = 3. Therefore, this option is not correct.

D) "The graph of f(x) = log_3(x) + c must intersect with the line x = c + 1 when y = 3."
If we set y = 3, we have 3 = log_3(x) + c.
Simplifying, we get log_3(x) = 3 - c.
This means that the point of intersection between the graph and the line x = c + 1 would occur when log_3(x) = 3 - c.
However, this does not necessarily happen at x = 3. Therefore, this option is not correct.

Based on our analysis, the correct choice is B) "The graph of f(x) = log_3(x) + c must intersect with the line y = c when x = 3."