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# Exponentials and Logarithms

When the exponential function was introduced, (for ) you saw that the function is increasing if and decreasing if . You can observe the monotonicity by plotting and .
> plot(exp(x),x=-1..1);
> plot(0.1^x,x=-1..1);
The logarithmic function was introduced for as the inverse of the exponential. The logarithm is therefore inreasing if and decreasing if . Indeed you can plot and .
> 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 logarithm of different bases by using the command. To get the animation to play, just click on the graph and click on the go button in the tool bar.
> 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);
From the monotonicuty properties you can see that for a fixed in the interval the logarithm increases with but for in the interval it decreases. As a result you can now compare logarithms with different bases without computing their values.

Here are some examples using Maple to solve logarithmic and exponential equations.

> solve(5^(x+1)+5^x+5^(x-1)=155,x);
> solve(log[5](2*x^2+2*x+5)=2,x);

Next: Inverse Functions Up: lab_template Previous: lab_template
Jane E Bouchard
2010-04-12