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Exponentials and Logarithms

When the exponential function tex2html_wrap_inline357 was introduced, (for tex2html_wrap_inline359 , b > 0) you saw that the function is increasing if b > 1 and decreasing if b < 1. You can observe the monotonicity by plotting tex2html_wrap_inline367 and tex2html_wrap_inline369 . Note that in Maple e is capitalized.

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

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

The logarithmic function tex2html_wrap_inline373 was introduced for tex2html_wrap_inline375 as the inverse of the exponential. The logarithm is therefore increasing if b > 1 and decreasing if 0 < b < 1. Indeed you can plot tex2html_wrap_inline381 and tex2html_wrap_inline383 . Note that the notation for tex2html_wrap_inline373 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 tex2html_wrap_inline373 with b, for a fixed x. You can use ``animate'' to plot the family of functions tex2html_wrap_inline373 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) tex2html_wrap_inline373 increases with b but for x in the interval tex2html_wrap_inline403 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.



  > 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));


next up previous
Next: Inverse functions Up: Background Previous: Background

Roxanne Tisch
Thu Apr 10 08:50:01 EDT 1997