Answers to Tutorial on Logarithmic Functions

Answers to the questions in tutorial (4) of Logarithmic functions are presented.

Question:
Investigate base B: set a=1, b=1, c=0 and d=0 using the scroll bar. Set B to values between 0 and 1 and to values greater than one, take note of the different graphs obtained and explain.

Answer:
For values of B between 0 and 1, the graph of f(x) = logB(x) decreases. For values of B greater than 1, the graph of f increases.


Question:
Investigate the effects of parameter a (vertical scaling) by setting B=e, b=1, c=0 and d=0.

Answer:
As a increases, the graph of f(x) = a*logB(x) is stretched vertically. As a decreases, the graph of f is compressed vertically


Question:
Investigate the effects of parameter b (horizontal scaling) by setting a=1, c=0, d=0 and B=e.

Answer:
As b increases, the graph of f(x) = logB(b*x) is compressed horizontally. As b decreases, the graph of f is stretched (expanded) horizontally.


Question:
Set B=e, a=1, b=1 and investigate the effects of c (horizontal shifting) and d (vertical translation).

Answer:
If we increase c from 0 to positive values, the graph of f(x) = logB(x + c) is shifted to the right. If we decrease c from 0 to negative values, the graph of f(x) = logB(x + c) is shifted to the left.
If we increase d from 0 to positive values, the graph is shifted up. If we decrease d from 0 to negative values, the graph is shifted down.




Question:
Set B, a, and d to some values and explain how parameters b and c affect the domain of the logarithmic function. Explain analytically.

Answer:
The domain of the logarithmic function given by f(x) = a*logB(b*x + c) + d is the set of all values of x that satisfy the condition


b*x + c > 0


Question:
What parameter(s) affect the x intercept? Is there always an x intercept? Explain analytically.

Answer:
The x intercept of the graph of the logarithmic function given by
f(x) = a*logB(b*x + c) + d is the value of x that satisfy the condition

a*logB(b*x + c) + d = 0
Solve the above equation for x (steps are shown below).
log
B(b*x + c) = -d/a
b*x + c =B
-d/a
x = [ B
-d/a - c ] / b
The graph of f will always have an x intercept (assuming a and b both not equal to zero).


Question:
What parameter(s) affect the y intercept? Is there always a y intercept? Explain analytically.
Answer:
The y intercept of the graph of the logarithmic function given by
y = f(0) = a*logB(c) + d.
The graph of f will have a y intercept only if c > 0.


Question:
What parameter(s) affect the vertical asymptote? Explain analytically.
Answer:
The vertical asymptote is a vertical line of the form x = k. It is found by solving the equation
b*x + c = 0
x = -c / b
The graph will always have a vertical asymptote (assuming b is not equal to zero).