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.

> plot(E^x,x=-1..1);

> plot((.1)^x,x=-1..1);

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.

> plot(log[10](x),x=0.1..10);

> plot(log[0.1](x),x=0.1..10);

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.

> with(plots):

> animate(log[b](x),x=0.1..10,b=1.1..10,frames=30);

> animate(log[b](x),x=0.1..10,b=0.01..0.1,frames=30);

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:

> solve(5^(x+1)+5^(x)+5^(x-1)=155,x);

> evalf(solve(3^x+3^(x+1)=108,x));

> solve(log[sqrt(x-1)](2*x^2+2*x+5)=4,x);

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.

Differentiate:

> diff(x^(x^x),x);

> diff(x^(ln(x)),x);

Use Maple to integrate:

> int(E^x/(5+(E^x)^2),x);

> int(sqrt(E^x-1),x=0..ln(2));

Thu Apr 10 08:50:01 EDT 1997