1000 decimal digits? how do I count them?

[Travis Gn]

I just started using Java, I completed my homework assignment in VB, but now I want to do it in java.
Problem: What is the largest value of n for which 2 to the nth power has fewer than 100 decimal digits. Also, fewer than 1000 decimal digits?
I don't know how to tell the length of a string, or a double, or what ever type I should be using for this problem?
I also don't know how to raise a number to a power? I guess just using a loop and multplying it again and again? It seems there should be a better way, but I can't find it.
The more help the better, any code examples are greatly appriciated. I seem to learn best from them.

[Gary Mann]

Check out the java.lang.Math class. It has a method for raising powers (pow) plus lots of other useful stuff.
I suspect your homework is a programming exercise rather than a Math problem, but I think there is a simple way to calculate this. It's a long time since I was at high school, but to determine "n", can't you just calc the log to base 2 of the target and take the integer part? e.g. log to base 2 of 100 is 6.629 so n = 6.
In Java, a method to do this would be:

public int calcHighestPowerOf2(double target) {
    return (int) (Math.log(target - 1) / Math.log(2));
}

This runs into trouble for very large numbers due to the size of a double. e.g. 1 followed by 1000 digits is too large. If the target is always a power of 10, then you could use:

public int calcHighestPowerOf2(int powerOf10) {
    return (int) (powerOf10 * Math.log(10) / Math.log(2));
}

[Travis Gn]

I took your advise about the math. I think that saved me lines of code and hours of headaches. Thank you. Since you were able to help with the math on that last one, I am working on another.
x^x = 10^100
and
x^x = 10^1000
I read over some stuff:
exp(ln(x)*ln(x) = 100*ln(10) ???
another thing I read talked about a "W" function? It lost me though.
I am very thankful for any help you provide.
Travis

[Gary Mann]

Sorry - none of this means anything to me. There is info on the net regarding the W function but that is advanced Mathematics. If that's what you're doing, a Java forum probably isn't the place to find answers. It may be time to check in with your teacher! Good Luck.