# Logarithmic Functions

An interactive applet is used to explore logarithmic functions and the properties of their graphs such domain, range, x and y intercepts and vertical asymptote.

Parameters included in the definition of the logarithmic function may be changed, using sliders, to investigate its properties. The continuous (small increments) changes of these parameters help in gaining a deep understanding of logarithmic functions.

The function to be explored has the form

f(x) = a*logB[ b (x+c) ] + d

a, b, c and d are coefficients and B is the base of the logarithm.

## Definition of Logarithmic Function

The logarithmic function is defined as the inverse of the exponential function.

For B > 0 and B not equal to 1,

y = Log Bx    is equivalent to    x = B y.

Note: The logarithm to the base e is written ln(x).

Example

1. f(x) = log2x
2. g(x) = log4x
3. h(x) = log0.5x

Logarithmic functions may be explored using either

### 2)A Java Applet: Interactive Tutorial (1)

Your browser is completely ignoring the <APPLET> tag!

1 - Click on the button above "click here to start" and maximize the window obtained.

2 - Use the sliders on the left side of the control panel of the applet to set a = 1, b = 1, c = 0, d = 0 and B = 2. These values define function f in part a) of the above example. Check few points on the graph such as log21 = 0, log22 = 1, log24 = 2. Use zoom in and out if necessary.

3 - Keep the same values for a, b, c and as above and set B = 4 to define function g in part b) above. Check few points such as log41 = 0, log44 = 1, log216 = 4, log464 = 3.

4 - Keep the same values for a, b, c and as above and set B = 0.5 to define function g in part c) above. Check few points such as log0.51 = 0, log0.52 = -1, log0.54 = -2, log0.58 = -3.

## Domain and Range of the Logarithmic Function

Let f(x) = logBx.

Since the exponential function is the inverse of the logarithmic function, the range of the logarithmic function is the domain of the exponential function which is the set of all real numbers.

The domain of the logarithmic function is the range of the exponential function which is given by the interval (0 , + infinity).

Interactive Tutorial (2)

1 - Use the sliders on the left side of the control panel of the applet to set a = 1, b = 1, c = 0, d = 0 and change base B. Observe the domain and range of the logarithmic function.

## Vertical Asymptote of the Logarithmic Function

logB0 is undefined. However it is possible to investigate the behavior of the graph the logarithmic function as x gets closer to zero from the right (x > 0).

Example

Let f(x) = log3x and find values f(x) as x gets closer to zero. Results are shown in the table below.

As x gets closer to zero, f(x) decreases without bound. The graph gets closer to the y axis (x = 0). The vertical line x = 0 is called the vertical asymptote.

Interactive Tutorial (3)

1 - Use the sliders on the left side of the control panel of the applet to set a = 1, b = 1, c = 0, d = 0 and change the base. Observe the behavior of the graph close to the y axis.

## Shifting, Scaling and Reflection of the graph of Logarithmic Functions

Interactive Tutorial (4)

1 - 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.

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

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

4 - set B=e, a=1, b=1 and investigate the effects of c (horizontal shifting) and d (vertical translation).

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

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

7 - What parameter(s) affect the y intercept? Is there always a y intercept? Explain analytically.

8 - What parameter(s) affect the vertical asymptote? Explain analytically.

More tutorials and self tests on logarithmic functions.