Logs by the fire

This problem is a nice introduction that will give you a feeling for how logs work and what that button on your calculator might be doing.

Here you have an opportunity to explore the proofs of the laws of logarithms.

Here you have an expression containing logs and factorials! What can you do with it?

Have you ever tried to work out the largest number that your calculator can cope with? What about your computer? Perhaps you tried using powers to make really large numbers. In this problem you will think about how much you can do to understand such numbers when your calculator is less than helpful!

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

Compares the size of functions f(n) for large values of n.

Solve these equations.