When the exponential function was introduced, (for , b > 0) you saw that the function is increasing if b > 1 and decreasing if b < 1. You can observe the monotonicity by plotting and . Note that in Maple e is capitalized.
The logarithmic function was introduced for as the inverse of the exponential. The logarithm is therefore increasing if b > 1 and decreasing if 0 < b < 1. Indeed you can plot and . Note that the notation for in Maple is: log[b]x.
From the monotonicity properties you can compare two logarithms having the same base, without computing their values:
We can also see what is the behavior of the with b, for a fixed x. You can use ``animate'' to plot the family of functions for different values of b.
In both animations you see that for a fixed x in the interval (0,1) increases with b but for x in the interval it decreases.
As a result you can now compare logarithms to different bases without computing their values:
Here are some examples how you can use Maple to solve logarithmic and exponential equations. Solve:
Observe that the command ``evalf'' makes Maple evaluate the result. Without it you get a result in terms of a logarithm.
You can also use Maple to differentiate and integrate exponential and logarithmic functions, as can be seen from the following examples.
Use Maple to integrate: