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Exponentials and Logarithms

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 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));
```

Next: Inverse functions Up: Background Previous: Background

Sean O Anderson
Tue Nov 28 15:50:23 EST 1995