Exponential and Logarithmic Functions
Introduction to exponential and logarithmic functions.
This applet shows exponential and/or logarithmic functions to the base as well as . The display of each can be turned on or off with the check boxes.
The value of is controlled by the slider and can be varied from -2 to 10. The increment is 0.1.
The exponential function is ( The base to the power x ).
The logarithmic function is the inverse of the exponent. If then .
When both exponential and logarithmic functions are shown the symmetry line is also shown.
To contrast these functions with a power the plot of can also be shown.
Answer these questions for both exponential and logarithmic functions
- For b > 1, What is the value of y for large negative x values?
- For b > 1, What is the value of y as x approaches zero?
- How do the curves for b=2 and b=1/2 compare?
- What is the domain ( possible x values ), including the dependance on b values ?
- What is the range ( possible y values ), including the dependance on b values ?
- What can you say about the power plot when b is not an integer?
- Why is the curve dotted for negative values of the base, b? Note: function values for integer x numbers are shown as dots.
Negative base
This is an advanced topic.
Odd roots of a negative number results in a negative number and even roots of a negative number results in an imaginary number. This means that odd roots gives a real number for negative values. The equivalent exponent rule indicates that if is odd a real number would result. Therefor, would have a real result whenever is a rational number with an odd denominator. It would also be a positive number if the numerator is even and a negative number if the numerator is odd. The final result is that the exponential function is not continuous because it jumps from positive to negative values and is missing irrational and rational numbers with even denominators.
The logarithmic function is obtained by switching the and values.