__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) = log_{B}(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*log_{B}(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) = log_{B}(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) = log_{B}(x + c) is shifted to the right. If we decrease c from 0 to negative values, the graph of f(x) = log_{B}(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*log_{B}(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*log_{B}(b*x + c) + d is the value of x that satisfy the condition

a*log_{B}(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*log_{B}(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).